Algebra - Elements of Combinatorics

Solve the equation: $\;$ $11 C^{x}_{3} = 24 C^{x+1}_{2}$, $\;\;$ $x \in N$

$11 C^{x}_{3} = 24 C^{x+1}_{2}$

i.e. $\;$ $\dfrac{11 \times x!}{3! \left(x - 3\right)!} = \dfrac{24 \times \left(x + 1\right)!}{2! \left(x + 1 - 2\right)!}$

i.e. $\;$ $\dfrac{11 \times x!}{3! \left(x - 3\right)!} = \dfrac{24 \times \left(x + 1\right)!}{2! \left(x - 1\right)!}$

i.e. $\;$ $\dfrac{11 \times x!}{ 3 \times 2! \left(x - 3\right)!} = \dfrac{24 \times \left(x + 1\right) x!}{2! \left(x - 1\right) \left(x - 2\right) \left(x - 3\right)!}$

i.e. $\;$ $\dfrac{11}{3} = \dfrac{24 \left(x + 1\right)}{\left(x - 1\right) \left(x - 2\right)}$

i.e. $\;$ $11 x^2 - 33x + 22 = 72x + 72$

i.e. $\;$ $11 x^2 - 105 x - 50 = 0$

i.e. $\;$ $\left(11x + 5\right) \left(x - 10\right) = 0$

i.e. $\;$ $11x + 5 = 0$ $\;$ or $\;$ $x - 10 = 0$

i.e. $\;$ $x = \dfrac{-5}{11}$ $\;$ or $x = 10$

$\because \;$ $x \in N$ $\implies$ $x = 10$ $\;$ is the required solution.

Algebra - Elements of Combinatorics

Solve the equation: $\;$ $C^{x}_3 + C^{x}_{4} = 11 \times C^{x+1}_{2}$, $\;\;$ $x \in N$

$C^{x}_3 + C^{x}_{4} = 11 \times C^{x+1}_{2}$

i.e. $\;$ $C^{x+1}_{4} = 11 \times C^{x+1}_{2}$ $\;\;\;$ $\left[\because \; C^{n}_{m} + C^{n}_{m + 1} = C^{n + 1}_{m + 1}\right]$

i.e. $\;$ $\dfrac{\left(x + 1\right)!}{4! \left(x + 1 - 4\right)!} = 11 \times \dfrac{\left(x + 1\right)!}{2! \left(x + 1 - 2\right)!}$

i.e. $\;$ $2! \left(x - 1\right)! = 11 \times 4! \left(x - 3\right)!$

i.e. $\;$ $2! \left(x - 1\right) \left(x - 2\right) \left(x - 3\right)! = 11 \times 4 \times 3 \times 2! \left(x - 3\right)!$

i.e. $\;$ $\left(x - 1\right) \left(x - 2\right) = 12 \times 11$

$\implies$ $x - 1 = 12$ $\;$ or $\;$ $x - 2 = 11$

$\implies$ $x = 13$

Algebra - Elements of Combinatorics

Solve the equation: $\;$ $\dfrac{P_{x + 3}}{P^{x}_{5} \times P_{x-5}} = 720$, $\;\;$ $x \in N$

$\dfrac{P_{x + 3}}{P^{x}_{5} \times P_{x-5}} = 720$

i.e. $\;$ $\dfrac{\left(x + 3\right)!}{\dfrac{x!}{\left(x - 5\right)!} \times \left(x - 5\right)!} = 720$

i.e. $\;$ $\dfrac{\left(x + 3\right)!}{x!} = 720$

i.e. $\;$ $\dfrac{\left(x + 3\right) \left(x + 2\right) \left(x + 1\right) x!}{x!} = 720$

i.e. $\;$ $\left(x + 3\right) \left(x + 2\right) \left(x + 1\right) = 10 \times 9 \times 8$

$\implies$ $x + 3 = 10$ $\;\;$ or $\;\;$ $x + 2 = 9$ $\;\;$ or $x + 1 = 8$

$\implies$ $x = 7$

Algebra - Elements of Combinatorics

Solve the equation: $\;$ $C^{n+1}_{m+1} : C^{n+1}_{m} : C^{n+1}_{m-1} = 5 : 5 : 3$

$C^{n+1}_{m+1} : C^{n+1}_{m} : C^{n+1}_{m-1} = 5 : 5 : 3$

i.e. $\;$ $C^{n+1}_{m+1} : C^{n+1}_{m} = 5 : 5$ $\;$ and $\;$ $C^{n+1}_{m} : C^{n+1}_{m-1} = 5 : 3$

Consider $\;\;\;$ $C^{n+1}_{m+1} : C^{n+1}_{m} = 5 : 5$

i.e. $\;$ $\dfrac{\left(n + 1\right)!}{\left(m + 1\right)! \left(n + 1 - m - 1\right)!} : \dfrac{\left(n + 1\right)!}{m! \left(n + 1 - m\right)!} = 1$

i.e. $\;$ $\dfrac{\left(n + 1\right)!}{\left(m + 1\right)! \left(n - m\right)!} \times \dfrac{m! \left(n + 1 - m\right)!}{\left(n + 1\right)!} = 1$

i.e. $\;$ $\dfrac{m! \left(n + 1 - m\right) \left(n - m\right)!}{\left(m + 1\right) m! \left(n - m\right)!} = 1$

i.e. $\;$ $n + 1 - m = m + 1$

i.e. $\;$ $n = 2m$ $\;\;\; \cdots \; (1)$

Consider $\;\;\;$ $C^{n+1}_{m} : C^{n+1}_{m-1} = 5 : 3$

In view of $(1)$ this becomes

$C^{2m + 1}_{m} : C^{2m + 1}_{m - 1} = 5 : 3$

i.e. $\;$ $\dfrac{\left(2m + 1\right)!}{m! \left(2m + 1 - m\right)!} : \dfrac{\left(2m + 1\right)!}{\left(m - 1\right)! \left(2m + 1 - m + 1\right)!} = \dfrac{5}{3}$

i.e. $\;$ $\dfrac{\left(m - 1\right)! \left(m + 2\right)!}{m! \left(m + 1\right)!} = \dfrac{5}{3}$

i.e. $\;$ $\dfrac{\left(m - 1\right)! \left(m + 2\right) \left(m + 1\right)!}{m \left(m -1\right)! \left(m + 1\right)!} = \dfrac{5}{3}$

i.e. $\;$ $\dfrac{m + 2}{m} = \dfrac{5}{3}$

i.e. $\;$ $3m + 6 = 5m$

i.e. $\;$ $2m = 6$ $\implies$ $m = 3$

Substituting the value of $m$ in $(1)$ gives

$n = 2m = 6$

Algebra - Elements of Combinatorics

Solve the equation: $\;$ $C^{x + 1}_{3} : C^{x}_{4} = 6 : 5$, $\;$ $x \in N$

$C^{x + 1}_{3} : C^{x}_{4} = 6 : 5$

i.e. $\;$ $\dfrac{\left(x + 1\right)!}{3! \left(x + 1 - 3\right)!} : \dfrac{x!}{4! \left(x - 4\right)!} = 6: 5$

i.e. $\;$ $\dfrac{\left(x + 1\right)!}{3! \left(x - 2\right)!} \times \dfrac{4! \left(x - 4\right)!}{x!} = \dfrac{6}{5}$

i.e. $\;$ $\dfrac{\left(x + 1\right) x!}{3! \left(x - 2\right) \left(x - 3\right) \left(x - 4\right)!} \times \dfrac{4 \times 3! \left(x - 4\right)!}{x!} = \dfrac{6}{5}$

i.e. $\;$ $\dfrac{4 \left(x + 1\right)}{\left(x - 2\right) \left(x - 3\right)}= \dfrac{6}{5}$

i.e. $\;$ $\dfrac{2 \left(x + 1\right)}{x^2 - 5x + 6} = \dfrac{3}{5}$

i.e. $\;$ $10x + 10 = 3x^2 - 15x + 18$

i.e. $\;$ $3x^2 - 25x + 8 = 0$

i.e. $\;$ $\left(x - 8\right) \left(3x - 1\right) = 0$

i.e. $\;$ $x = 8$ $\;$ or $\;$ $x = \dfrac{1}{3}$

$\because \;$ $x \in N$ $\;\;\;$ [given]

$\implies$ $x = 8$ $\;$ is the required solution.

Algebra - Elements of Combinatorics

Solve the equation: $\;$ $C^{x+1}_{x-4} = \dfrac{7}{15} P^{x+1}_{3}$, $\;$ $x \in N$

$C^{x+1}_{x-4} = \dfrac{7}{15} P^{x+1}_{3}$

i.e. $\;$ $\dfrac{\left(x + 1\right)!}{\left(x - 4\right)! \left(x + 1 - x + 4\right)!} = \dfrac{7}{15} \times \dfrac{\left(x + 1\right)!}{\left(x + 1 - 3\right)!}$

i.e. $\;$ $\dfrac{15}{\left(x - 4\right)! \times 5!} = \dfrac{7}{\left(x - 2\right)!}$

i.e. $\;$ $\dfrac{15}{\left(x - 4\right)! \times 5 \times 4 \times 3 \times 2} = \dfrac{7}{\left(x - 2\right) \left(x - 3\right) \left(x - 4\right)!}$

i.e. $\;$ $\dfrac{1}{8} = \dfrac{7}{\left(x - 2\right) \left(x - 3\right)}$

i.e. $\;$ $\left(x - 2\right) \left(x - 3\right) = 8 \times 7$

$\implies$ $x - 2 = 8$ $\;$ or $\;$ $x - 3 = 7$

$\implies$ $x = 10$

Algebra - Elements of Combinatorics

Solve the equation: $\;$ $P^{x + 1}_{3} + C^{x + 1}_{x - 1} = 14 \left(x + 1\right)$, $\;$ $x \in N$

$P^{x + 1}_{3} + C^{x + 1}_{x - 1} = 14 \left(x + 1\right)$

i.e. $\;$ $\dfrac{\left(x + 1\right)!}{\left(x + 1 - 3\right)!} + \dfrac{\left(x + 1\right)!}{\left(x - 1\right)! \left(x + 1 - x + 1\right)!} = 14 \left(x + 1\right)$

i.e. $\;$ $\dfrac{\left(x + 1\right)!}{\left(x - 2\right)!} + \dfrac{\left(x + 1\right)!}{\left(x - 1\right)! 2!} = 14 \left(x + 1\right)$

i.e. $\;$ $\dfrac{\left(x + 1\right) x \left(x - 1\right) \left(x - 2\right)!}{\left(x - 2\right)!} + \dfrac{\left(x + 1\right) x \left(x - 1\right)!}{2 \left(x - 1\right)!} = 14\left(x + 1\right)$

i.e. $\;$ $x \left(x + 1\right) \left(x - 1\right) + \dfrac{x \left(x + 1\right)}{2} = 14\left(x + 1\right)$

i.e. $\;$ $x \left(x - 1 + \dfrac{1}{2}\right) = 14$

i.e. $\;$ $\dfrac{x \left(2x - 1\right)}{2} = 14$

i.e. $\;$ $2x^2 - x - 28 = 0$

i.e. $\;$ $2 \left(x - 4\right) \left(x + 3.5\right) = 0$

$\implies$ $x = 4$, $\;$ or $\;$ $x = -3.5$

$\because \;$ $x \in N$ $\;\;\;$ [given]

$\implies$ $x = 4$ $\;$ is the required solution.

Algebra - Elements of Combinatorics

Solve the equation: $\;$ $12 C^{x}_{1} + C^{x + 4}_{2} = 162$, $\;$ $x \in N$

$12 C^{x}_{1} + C^{x + 4}_{2} = 162$

i.e. $\;$ $\dfrac{12 \times x!}{1! \left(x - 1\right)!} + \dfrac{\left(x + 4\right)!}{2! \left(x + 4 - 2\right)!} = 162$

i.e. $\;$ $\dfrac{12 x \left(x - 1\right)!}{\left(x - 1\right)!} + \dfrac{\left(x + 4\right)!}{2 \left(x + 2\right)!} = 162$

i.e. $\;$ $12 x + \dfrac{\left(x + 4\right) \left(x + 3\right) \left(x + 2\right)!}{2 \left(x + 2\right)!} = 162$

i.e. $\;$ $12x + \dfrac{\left(x + 4\right) \left(x + 3\right)}{2} = 162$

i.e. $\;$ $24x + x^2 + 7x + 12 = 324$

i.e. $\;$ $x^2 + 31x - 312 = 0$

i.e. $\;$ $\left(x + 39\right) \left(x - 8\right) = 0$

$\implies$ $x = -39$ $\;$ or $\;$ $x = 8$

$\because \;$ $x \in N$ $\;\;$ [given]

$\implies$ $x = 8$ $\;$ is the required solution.

Algebra - Elements of Combinatorics

Solve the equation: $\;$ $P^{x-1}_{2} - C^{x}_{1} = 79$, $\;$ $x \in N$

$P^{x-1}_{2} - C^{x}_{1} = 79$

i.e. $\;$ $\dfrac{\left(x - 1\right)!}{\left(x - 1 - 2\right)!} - \dfrac{x!}{1! \left(x - 1\right)!} = 79$

i.e. $\;$ $\dfrac{\left(x - 1\right)!}{\left(x - 3\right)!} - \dfrac{x!}{1 \times \left(x - 1\right)!} = 79$

i.e. $\;$ $\dfrac{\left(x - 1\right) \left(x - 2\right) \left(x - 3\right)!}{\left(x - 3\right)!} - \dfrac{x \left(x - 1\right)!}{\left(x - 1\right)!} = 79$

i.e. $\;$ $\left(x - 1\right) \left(x - 2\right) - x = 79$

i.e. $\;$ $x^2 - 3x + 2 - x = 79$

i.e. $\;$ $x^2 - 4x -77 = 0$

i.e. $\;$ $x^2 -11x + 7x - 77 = 0$

i.e. $\;$ $x \left(x - 11\right) + 7 \left(x - 11\right) = 0$

i.e. $\;$ $\left(x - 11\right) \left(x + 7\right) = 0$

$\implies$ $x = 11$ $\;$ or $\;$ $x = -7$

$\because \;$ $x \in N$ $\;\;$ [given]

$\implies$ $x = 11$ $\;$ is the required solution.

Algebra - Elements of Combinatorics

Solve the equation: $\;$ $\dfrac{C^{2x}_{x+1}}{C^{2x+1}_{x-1}} = \dfrac{2}{3}$, $\;$ $x \in N$

$\dfrac{C^{2x}_{x+1}}{C^{2x+1}_{x-1}} = \dfrac{2}{3}$

i.e. $\;$ $\dfrac{2x!}{\left(x + 1\right)! \left(2x - x - 1\right)!} \div \dfrac{\left(2x + 1\right)!}{\left(x - 1\right)! \left(2x + 1 - x + 1\right)!} = \dfrac{2}{3}$

i.e. $\;$ $\dfrac{2x!}{\left(x + 1\right)! \left(x - 1\right)!} \times \dfrac{\left(x - 1\right)! \left(x + 2\right)!}{\left(2x + 1\right)!} = \dfrac{2}{3}$

i.e. $\;$ $\dfrac{2x!}{\left(x + 1\right)!} \times \dfrac{\left(x + 2\right) \left(x + 1\right)!}{\left(2x + 1\right) \left(2x\right)!} = \dfrac{2}{3}$

i.e. $\;$ $\dfrac{x + 2}{2x + 1} = \dfrac{2}{3}$

i.e. $\;$ $3x + 6 = 4x + 2$

i.e. $\;$ $x = 4$

Algebra - Algebraic Expressions

Prove the identity $\;$ $\dfrac{b^2 - 3b - \left(b - 1\right) \sqrt{b^2 - 4} + 2}{b^2 + 3b - \left(b + 1\right) \sqrt{b^2 - 4} + 2} \sqrt{\dfrac{b + 2}{b - 2}} = \dfrac{1 - b}{1 + b}$

LHS $= \dfrac{b^2 - 3b - \left(b - 1\right) \sqrt{b^2 - 4} + 2}{b^2 + 3b - \left(b + 1\right) \sqrt{b^2 - 4} + 2} \sqrt{\dfrac{b + 2}{b - 2}}$

$= \dfrac{b^2 - b - 2b - \left(b - 1\right) \sqrt{b^2 - 4} + 2}{b^2 + b + 2b - \left(b + 1\right) \sqrt{b^2 - 4} + 2} \times \dfrac{\sqrt{b + 2}}{\sqrt{b - 2}}$

$= \dfrac{b \left(b - 1\right) - 2 \left(b - 1\right) - \left(b - 1\right) \sqrt{b^2 - 4}}{b \left(b + 1\right) + 2 \left(b + 1\right) - \left(b + 1\right) \sqrt{b^2 - 4}} \times \dfrac{\sqrt{b + 2}}{\sqrt{b - 2}}$

$= \dfrac{\left(b - 1\right) \left[b - 2 - \sqrt{b^2 - 4}\right]}{\left(b + 1\right) \left[b + 2 - \sqrt{b^2 - 4}\right]} \times \dfrac{\sqrt{b + 2}}{\sqrt{b - 2}}$

$= \dfrac{\left(b - 1\right) \left[\sqrt{b - 2} \times \sqrt{b - 2} - \sqrt{b + 2} \times \sqrt{b - 2}\right]}{\left(b + 1\right) \left[\sqrt{b + 2} \times \sqrt{b + 2} - \sqrt{b + 2} \times \sqrt{b - 2}\right]} \times \dfrac{\sqrt{b + 2}}{\sqrt{b - 2}}$

$= \dfrac{\left(b - 1\right) \left(\sqrt{b - 2}\right) \left(\sqrt{b - 2} - \sqrt{b + 2}\right)}{\left(b + 1\right) \left(\sqrt{b + 2}\right) \left(\sqrt{b + 2} - \sqrt{b - 2}\right)} \times \dfrac{\sqrt{b + 2}}{\sqrt{b - 2}}$

$= \dfrac{-\left(b - 1\right)}{b + 1}$

$= \dfrac{1 - b}{1 + b}$

$= RHS$

Algebra - Algebraic Expressions

Simplify: $\;$ $\left[\dfrac{a^2}{a + b} - \dfrac{a^3}{a^2 + 2ab + b^2}\right] : \left[\dfrac{a}{a + b} - \dfrac{a^2}{a^2 - b^2}\right]$ $\;$ and calculate it for $\;$ $a = - 2.5$, $\;$ $b = 0.5$.

$\left[\dfrac{a^2}{a + b} - \dfrac{a^3}{a^2 + 2ab + b^2}\right] : \left[\dfrac{a}{a + b} - \dfrac{a^2}{a^2 - b^2}\right]$

$= \left[\dfrac{a^2}{a + b} - \dfrac{a^3}{\left(a + b\right)^2}\right] : \left[\dfrac{a}{a + b} - \dfrac{a^2}{\left(a + b\right) \left(a - b\right)}\right]$

$= \left[\dfrac{a^2}{a + b} \left(1 - \dfrac{a}{a + b}\right)\right] : \left[\dfrac{a}{a + b} \left(1 - \dfrac{a}{a - b}\right)\right]$

$= \left[\dfrac{a^2 b}{\left(a + b\right)^2}\right] : \left[\dfrac{-ab}{\left(a + b\right) \left(a - b\right)}\right]$

$= \dfrac{a^2 b}{\left(a + b\right)^2} \times \dfrac{\left(a + b\right) \left(a - b\right)}{\left(-ab\right)}$

$= \dfrac{a \left(b - a\right)}{a + b}$

When $\;$ $a = -2.5$, $\;$ $b = 0.5$, $\;$ the given expression becomes

$\dfrac{-2.5 \times \left(0.5 + 2.5\right)}{-2.5 + 0.5}$

$= \dfrac{-2.5 \times 3}{-2}$

$= \dfrac{7.5}{2} = \dfrac{15}{4}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\dfrac{1 - a^{\frac{-1}{2}}}{1 + a^{\frac{1}{2}}} - \dfrac{a^{\frac{1}{2}} - a^{\frac{-1}{2}}}{a - 1}$ $\;$ and calculate it for $\;$ $a = 5$.

$\dfrac{1 - a^{\frac{-1}{2}}}{1 + a^{\frac{1}{2}}} - \dfrac{a^{\frac{1}{2}} - a^{\frac{-1}{2}}}{a - 1}$

$= \dfrac{1 - \dfrac{1}{\sqrt{a}}}{\sqrt{a} + 1} - \dfrac{\sqrt{a} - \dfrac{1}{\sqrt{a}}}{a - 1}$

$= \dfrac{\sqrt{a} - 1}{\sqrt{a} \left(\sqrt{a} + 1\right)} - \dfrac{a - 1}{\sqrt{a} \left(a - 1\right)}$

$= \dfrac{\sqrt{a} - 1}{\sqrt{a} \left(\sqrt{a} + 1\right)} - \dfrac{1}{\sqrt{a}}$

$= \dfrac{\sqrt{a} - 1 - \left(\sqrt{a} + 1\right)}{\sqrt{a} \left(\sqrt{a} + 1\right)}$

$= \dfrac{-2}{a + \sqrt{a}}$

When $\;$ $a = 5$, $\;$ the given expression becomes

$\dfrac{-2}{5 + \sqrt{5}}$

$= \dfrac{-2 \left(5 - \sqrt{5}\right)}{\left(5 + \sqrt{5}\right) \left(5 - \sqrt{5}\right)}$

$= \dfrac{-2 \left(5 - \sqrt{5}\right)}{20}$

$= \dfrac{\sqrt{5} - 5}{10}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\dfrac{\left(a^{-1} + b^{-1}\right) \left(a + b\right)^{-1}}{\sqrt[6]{a^{4} \sqrt[5]{a^{-2}}}}$

$\dfrac{\left(a^{-1} + b^{-1}\right) \left(a + b\right)^{-1}}{\sqrt[6]{a^{4} \sqrt[5]{a^{-2}}}}$

$= \dfrac{\left(\dfrac{1}{a} + \dfrac{1}{b}\right) \left(\dfrac{1}{a + b}\right)}{\left[a^4 \left(a^{-2}\right)^\frac{1}{5}\right]^{\frac{1}{6}}}$

$= \dfrac{a + b}{ab \left(a + b\right) \left[a^4 \times a^{\frac{-2}{5}}\right]^{\frac{1}{6}}}$

$= \dfrac{1}{ab \left(a^{\frac{18}{5}}\right)^{\frac{1}{6}}}$

$= \dfrac{1}{a b \times a^{\frac{3}{5}}}$

$= \dfrac{1}{a^{\frac{8}{5}} b}$

$= \dfrac{1}{b \sqrt[5]{a^8}}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\dfrac{\sqrt[3]{a + \sqrt{2 - a^2}} \sqrt[6]{1 - a \sqrt{2 - a^2}}}{\sqrt[3]{1 - a^2}}$, $\;\;$ $\left|a\right| < 1$

$\dfrac{\sqrt[3]{a + \sqrt{2 - a^2}} \sqrt[6]{1 - a \sqrt{2 - a^2}}}{\sqrt[3]{1 - a^2}}$

$= \dfrac{\left[a + \sqrt{2 - a^2}\right]^{\frac{1}{3}} \left[1 - a \sqrt{2 - a^2}\right]^{\frac{1}{6}}}{\left[1 - a^2\right]^{\frac{1}{3}}}$

$= \dfrac{\left[\left(a + \sqrt{2 - a^2}\right)^{\frac{1}{6}}\right]^2 \left(1 - a \sqrt{2 - a^2}\right)^{\frac{1}{6}}}{\left(1 - a^2\right)^{\frac{1}{3}}}$

$= \dfrac{\left[\left(a + \sqrt{2 - a^2}\right)^2\right]^{\frac{1}{6}} \left(1 - a \sqrt{2 - a^2}\right)^{\frac{1}{6}}}{\left(1 - a^2\right)^{\frac{1}{3}}}$

$= \dfrac{\left(a^2 + 2a \sqrt{2 - a^2} + 2 - a^2\right)^{\frac{1}{6}} \left(1 - a\sqrt{2 - a^2}\right)^{\frac{1}{6}}}{\left(1 - a^2\right)^{\frac{1}{3}}}$

$= \dfrac{2^{\frac{1}{6}} \left(1 + a \sqrt{2 - a^2}\right)^{\frac{1}{6}} \left(1 - a \sqrt{2 - a^2}\right)^{\frac{1}{6}}}{\left(1 - a^2\right)^{\frac{1}{3}}}$

$= \dfrac{2^{\frac{1}{6}} \left[\left(1 + a \sqrt{2 - a^2}\right) \left(1 - a \sqrt{2 - a^2}\right)\right]^{\frac{1}{6}}}{\left(1 - a^2\right)^{\frac{1}{3}}}$

$= \dfrac{2^{\frac{1}{6}} \left[1 - a^2 \left(2 - a^2\right)\right]^{\frac{1}{6}}}{\left(1 - a^2\right)^{\frac{1}{3}}}$

$= \dfrac{2^{\frac{1}{6}} \left[1 - 2a^2 + a^4\right]^{\frac{1}{6}}}{\left(1 - a^2\right)^{\frac{1}{3}}}$

$= \dfrac{2^{\frac{1}{6}} \left[\left(1 - a^2\right)^2\right]^{\frac{1}{6}}}{\left(1 - a^2\right)^{\frac{1}{3}}}$

$= \dfrac{2^{\frac{1}{6}} \left(1 - a^2\right)^{\frac{1}{3}}}{\left(1 - a^2\right)^{\frac{1}{3}}}$

$= 2^{\frac{1}{6}} = \sqrt[6]{2}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\left[\left(a - b\right) \sqrt{\dfrac{a + b}{a - b}} + a - b\right] \left(a - b\right) \left(\sqrt{\dfrac{a + b}{a - b}} - 1\right)$, $\;\;$ $a + b < 0$, $\;$ $a - b < 0$

$\left[\left(a - b\right) \sqrt{\dfrac{a + b}{a - b}} + a - b\right] \left(a - b\right) \left(\sqrt{\dfrac{a + b}{a - b}} - 1\right)$

$= \left[\dfrac{\left(a - b\right) \sqrt{a + b} + \left(a - b\right) \sqrt{a - b}}{\sqrt{a - b}}\right] \left(a - b\right) \left(\dfrac{\sqrt{a + b} - \sqrt{a - b}}{\sqrt{a - b}}\right)$

$= \dfrac{\left(a - b\right) \left(\sqrt{a + b} + \sqrt{a - b}\right)}{a - b} \times \left(a - b\right) \times \left(\sqrt{a + b} - \sqrt{a - b}\right)$

$= \left(a - b\right) \left(a + b - a + b\right)$

$= 2b \left(a - b\right)$

Algebra - Algebraic Expressions

Simplify: $\;$ $\dfrac{2a \sqrt{1 + x^2}}{x + \sqrt{1 + x^2}}$ $\;$ when $\;$ $x = \dfrac{1}{2} \left(\sqrt{\dfrac{a}{b}} - \sqrt{\dfrac{b}{a}}\right)$, $\;\;$ $a > 0, \; b > 0$

Given: $\;\;\;$ $x = \dfrac{1}{2} \left(\sqrt{\dfrac{a}{b}} - \sqrt{\dfrac{b}{a}}\right) = \dfrac{a - b}{2 \sqrt{ab}}$

Value of $\;$ $\sqrt{1 + x^2}$ $\;$ when $\;$ $x = \dfrac{1}{2} \left(\sqrt{\dfrac{a}{b}} - \sqrt{\dfrac{b}{a}}\right)$ $\;$ is

$\sqrt{1 + \left(\dfrac{a - b}{2ab}\right)^2}$

$= \sqrt{1 + \dfrac{a^2 - 2ab + b^2}{4ab}}$

$= \sqrt{\dfrac{4ab + a^2 - 2ab + b^2}{4ab}}$

$= \sqrt{\dfrac{a^2 + 2ab + b^2}{4ab}}$

$= \sqrt{\dfrac{\left(a + b\right)^2}{4ab}}$

$= \dfrac{a + b}{2 \sqrt{ab}}$

Value of $\;$ $x + \sqrt{1 + x^2}$ $\;$ when $\;$ $x = \dfrac{1}{2} \left(\sqrt{\dfrac{a}{b}} - \sqrt{\dfrac{b}{a}}\right)$ $\;$ is

$\dfrac{a - b}{2 \sqrt{ab}} + \dfrac{a + b}{2 \sqrt{ab}}$

$= \dfrac{2a}{2 \sqrt{ab}}$

$= \dfrac{a}{\sqrt{ab}}$

$\therefore \;$ Value of given expression $\;$ $\dfrac{2a \sqrt{1 + x^2}}{x + \sqrt{1 + x^2}}$ $\;$ when $\;$ $x = \dfrac{1}{2} \left(\sqrt{\dfrac{a}{b}} - \sqrt{\dfrac{b}{a}}\right)$ $\;$ is

$\dfrac{2a \left(\dfrac{a + b}{2 \sqrt{ab}}\right)}{\dfrac{a}{\sqrt{ab}}}$

$= a + b$

Algebra - Algebraic Expressions

Simplify: $\;$ $\left[\dfrac{1 + \sqrt{1 - x}}{1 - x + \sqrt{1 - x}} + \dfrac{1 - \sqrt{1 + x}}{1 + x - \sqrt{1 + x}}\right]^2 \left(\dfrac{x^2 - 1}{2}\right) + 1$, $\;\;$ $0 < x < 1$

$\left[\dfrac{1 + \sqrt{1 - x}}{1 - x + \sqrt{1 - x}} + \dfrac{1 - \sqrt{1 + x}}{1 + x - \sqrt{1 + x}}\right]^2 \left(\dfrac{x^2 - 1}{2}\right) + 1$, $\;\;$ $0 < x < 1$

$= \left[\dfrac{1 + \sqrt{1 - x}}{\sqrt{1 - x} \left(\sqrt{1 - x} + 1\right)} + \dfrac{1 - \sqrt{1 + x}}{\sqrt{1 + x} \left(\sqrt{1 + x} - 1\right)}\right]^2 \left(\dfrac{x^2 - 1}{2}\right) + 1$

$= \left[\dfrac{1}{\sqrt{1 - x}} - \dfrac{1}{\sqrt{1 + x}}\right]^2 \left(\dfrac{x^2 - 1}{2}\right) + 1$

$= \left[\dfrac{\sqrt{1 + x} - \sqrt{1 - x}}{\left(1 - x\right) \left(\sqrt{1 + x}\right)}\right]^2 \left(\dfrac{x^2 - 1}{2}\right) + 1$

$= \left[\dfrac{\sqrt{1 + x} - \sqrt{1 - x}}{\sqrt{1 - x^2}}\right]^2 \left(\dfrac{x^2 - 1}{2}\right) + 1$

$= \left[\dfrac{1 + x + 1 - x - 2 \sqrt{\left(1 + x\right) \left(1 - x\right)}}{1 - x^2}\right] \left(\dfrac{x^2 - 1}{2}\right) + 1$

$= \left[\dfrac{2 - 2 \sqrt{1 - x^2}}{1 - x^2}\right] \left(\dfrac{x^2 - 1}{2}\right) + 1$

$= \sqrt{1 - x^2} - 1 + 1$

$= \sqrt{1 - x^2}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\dfrac{\sqrt{a^2 - 2ab + b^2}}{\sqrt{a^2 + 2ab + b^2}} + \dfrac{2b}{a + b}$, $\;$ $0 < a < b$

$\dfrac{\sqrt{a^2 - 2ab + b^2}}{\sqrt{a^2 + 2ab + b^2}} + \dfrac{2b}{a + b}$

$= \dfrac{\sqrt{\left(a - b\right)^2}}{\sqrt{\left(a + b\right)^2}} + \dfrac{2b}{a + b}$

$= \dfrac{a - b}{a + b} + \dfrac{2b}{a + b}$

$= \dfrac{a + b}{a + b}$

$= 1$

Algebra - Algebraic Expressions

Simplify: $\;$ $\left[\dfrac{\left(\sqrt[3]{x} - \sqrt[3]{a}\right)^3 + \left(2x + a\right)}{\left(\sqrt[3]{x} - \sqrt[3]{a}\right)^3 - \left(x + 2a\right)}\right]^3 + \left[\sqrt{\left|a^3 - 3 a^2 x + 3 a x^2 - x^3\right|^{\frac{2}{3}}} : a\right]$ $\;\;$ for $x > a$

$\left[\dfrac{\left(\sqrt[3]{x} - \sqrt[3]{a}\right)^3 + \left(2x + a\right)}{\left(\sqrt[3]{x} - \sqrt[3]{a}\right)^3 - \left(x + 2a\right)}\right]^3 + \left[\sqrt{\left|a^3 - 3 a^2 x + 3 a x^2 - x^3\right|^{\frac{2}{3}}} : a\right]$ $\;\;\; \cdots \; (1)$

Consider the expression $\;\;$ $\left[\dfrac{\left(\sqrt[3]{x} - \sqrt[3]{a}\right)^3 + \left(2x + a\right)}{\left(\sqrt[3]{x} - \sqrt[3]{a}\right)^3 - \left(x + 2a\right)}\right]^3$

$= \left[\dfrac{x - 3 x^{\frac{2}{3}} a^{\frac{1}{3}} + 3 x^{\frac{1}{3}} a^{\frac{2}{3}} - a + 2x + a}{x - 3 x^{\frac{2}{3}} a^{\frac{1}{3}} + 3 x^{\frac{1}{3}} a^{\frac{2}{3}} - a - x - 2a}\right]^3$

$= \left[\dfrac{3x - 3 x^{\frac{2}{3}} a^{\frac{1}{3}} + 3 x^{\frac{1}{3}} a^{\frac{2}{3}}}{-3a - 3 x^{\frac{2}{3}} a^{\frac{1}{3}} + 3 x^{\frac{1}{3}} a^{\frac{2}{3}}}\right]^3$

$= \left[\dfrac{x^{\frac{1}{3}} \left(x^{\frac{2}{3}} - x^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right)}{-a^{\frac{1}{3}} \left(a^{\frac{2}{3}} + x^{\frac{2}{3}} - x^{\frac{1}{3}} a^{\frac{1}{3}}\right)}\right]^3$

$= \left[\left(\dfrac{-x}{a}\right)^{\frac{1}{3}}\right]^3$

$= \dfrac{-x}{a}$ $\;\;\; \cdots \; (2)$

Consider the expression $\;\;$ $\sqrt{\left|a^3 - 3 a^2 x + 3 a x^2 - x^3\right|^{\frac{2}{3}}}$

$= \sqrt{\left|\left(a - x\right)^3\right|^{\frac{2}{3}}}$

$= \sqrt{\left(x - a\right)^2}$ $\;\;$ for $\;$ $x > a$

$= x - a$ $\;\;\; \cdots \; (3)$

$\therefore \;$ In view of expressions $(2)$ and $(3)$, expression $(1)$ becomes

$\dfrac{-x}{a} + \dfrac{x - a}{a}$

$= \dfrac{-a}{a}$

$= -1$

Algebra - Algebraic Expressions

Simplify: $\;$ $\dfrac{1}{a^2} \sqrt{\left(a^6 + \dfrac{3 a^4}{b^{-2}} + \dfrac{a^2 b^4}{3^{-1}} + \dfrac{1}{b^{-6}}\right)^{\frac{2}{3}}} + \left[\dfrac{\left(b^{\frac{2}{3}} - a^{\frac{2}{3}}\right)^3 - 2a^2 - b^2}{a^2 + \left(b^{\frac{2}{3}} - a^{\frac{2}{3}}\right)^3 + 2 b^2}\right]^{-3}$

$\dfrac{1}{a^2} \sqrt{\left(a^6 + \dfrac{3 a^4}{b^{-2}} + \dfrac{a^2 b^4}{3^{-1}} + \dfrac{1}{b^{-6}}\right)^{\frac{2}{3}}} + \left[\dfrac{\left(b^{\frac{2}{3}} - a^{\frac{2}{3}}\right)^3 - 2a^2 - b^2}{a^2 + \left(b^{\frac{2}{3}} - a^{\frac{2}{3}}\right)^3 + 2 b^2}\right]^{-3}$ $\;\;\; \cdots \; (1)$

Consider the expression $\;\;$ $\dfrac{1}{a^2} \sqrt{\left(a^6 + \dfrac{3 a^4}{b^{-2}} + \dfrac{a^2 b^4}{3^{-1}} + \dfrac{1}{b^{-6}}\right)^{\frac{2}{3}}}$

$= \dfrac{1}{a^2} \left[\left(a^6 + 3 a^4 b^2 + 3 a^2 b^4 + b^6\right)^{\frac{2}{3}}\right]^{\frac{1}{2}}$

$= \dfrac{1}{a^2} \left[\left(a^2\right)^3 + 3 \left(a^2\right)^2 b^2 + 3 a^2 \left(b^2\right)^2 + \left(b^2\right)^3\right]^{\frac{1}{3}}$

$= \dfrac{1}{a^2} \left[\left(a^2 + b^2\right)^3\right]^{\frac{1}{3}}$

$= \dfrac{a^2 + b^2}{a^2}$

$= 1 + \dfrac{b^2}{a^2}$ $\;\;\; \cdots \; (2)$

Consider the expression $\;\;$ $\left[\dfrac{\left(b^{\frac{2}{3}} - a^{\frac{2}{3}}\right)^3 - 2a^2 - b^2}{a^2 + \left(b^{\frac{2}{3}} - a^{\frac{2}{3}}\right)^3 + 2 b^2}\right]^{-3}$

$= \left[\dfrac{\left(b^{\frac{2}{3}}\right)^3 - 3 \left(b^{\frac{2}{3}}\right)^2 a^{\frac{2}{3}} + 3 b^{\frac{2}{3}} \left(a^{\frac{2}{3}}\right)^2 - \left(a^{\frac{2}{3}}\right)^3 - 2a^2 - b^2}{a^2 + \left(b^{\frac{2}{3}}\right)^3 - 3 \left(b^{\frac{2}{3}}\right)^2 a^{\frac{2}{3}} + 3 b^{\frac{2}{3}} \left(a^{\frac{2}{3}}\right)^2 - \left(a^{\frac{2}{3}}\right)^3 + 2b^2} \right]^{-3}$

$= \left[\dfrac{b^2 - 3 b^{\frac{4}{3}} a^{\frac{2}{3}} + 3 b^{\frac{2}{3}} a^{\frac{4}{3}} - a^2 - 2a^2 - b^2}{a^2 + b^2 - 3 b^{\frac{4}{3}} a^{\frac{2}{3}} + 3 b^{\frac{2}{3}} a^{\frac{4}{3}} - a^2 + 2b^2}\right]^{-3}$

$= \left[\dfrac{3b^2 - 3 b^{\frac{4}{3}} a^{\frac{2}{3}} + 3 b^{\frac{2}{3}} a^{\frac{4}{3}}}{-3a^2 - 3b^{\frac{4}{3}} a^{\frac{2}{3}} + 3 b^{\frac{2}{3}} a^{\frac{4}{3}}}\right]^3$

$= \left[\dfrac{b^2 - a^{\frac{2}{3}} b^{\frac{2}{3}} \left(b^{\frac{2}{3}} - a^{\frac{2}{3}}\right)}{-a^2 + a^{\frac{2}{3}} b^{\frac{2}{3}} \left(a^{\frac{2}{3}} - b^{\frac{2}{3}}\right)}\right]^3$

$= \left[\dfrac{b^{\frac{2}{3}} \left(b^{\frac{4}{3}} - a^{\frac{2}{3}} b^{\frac{2}{3}} + a^{\frac{4}{3}}\right)}{-a^{\frac{2}{3}} \left(a^{\frac{4}{3}} - a^{\frac{2}{3}} b^{\frac{2}{3}} + b^{\frac{2}{3}}\right)}\right]^3$

$= \left[\dfrac{- b^{\frac{2}{3}}}{a^{\frac{2}{3}}}\right]^3$

$= \dfrac{- b^2}{a^2}$ $\;\;\; \cdots \; (3)$

In view of expressions $(2)$ and $(3)$, expression $(1)$ becomes

$1 + \dfrac{b^2}{a^2} - \dfrac{b^2}{a^2}$

$= 1$

Algebra - Algebraic Expressions

Simplify: $\;$ $\dfrac{2a \sqrt[3]{a b^2} - a \sqrt[6]{a b^5} - ab}{\sqrt[3]{a^2 b} - \sqrt{ab}} - 2^{\left(1 + 2 \log_8 a + \log_8 b\right)}$

$\dfrac{2a \sqrt[3]{a b^2} - a \sqrt[6]{a b^5} - ab}{\sqrt[3]{a^2 b} - \sqrt{ab}} - 2^{\left(1 + 2 \log_8 a + \log_8 b\right)}$

$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}} - 2^{\left(\log_8 8 + \log_8 a^2 + \log_8 b\right)}$

$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}} - 2^{\log_8 8a^2b}$

$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}} - 2^{\frac{\log_2 8a^2b}{\log_2 8}}$

$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}} - 2^{\frac{\log_2 8a^2b}{\log_2 2^3}}$

$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}} - 2^{\frac{\log_2 8a^2b}{3 \log_2 2}}$

$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}} - 2^{\frac{1}{3} \log_2 8a^2b}$

$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}} - 2^{\left(\log_2 8a^2b\right)^{\frac{1}{3}}}$

$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}} - \left(8a^2b\right)^{\frac{1}{3}}$

$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}} - 2 a^{\frac{2}{3} b^{\frac{1}{3}}}$

$= \dfrac{2a^{\frac{4}{3}} b^{\frac{2}{3}} - a^{\frac{7}{6}} b^{\frac{5}{6}} - ab - 2 a^{\frac{4}{3}} b^{\frac{2}{3}} + 2 a^{\frac{7}{6}} b^{\frac{5}{6}}}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}}$

$= \dfrac{a^{\frac{7}{6}} b^{\frac{5}{6}} - ab}{a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{2}} b^{\frac{1}{2}}}$

$= \dfrac{ab \left(a^{\frac{1}{6}} b^{\frac{-1}{6}} - 1\right)}{a^{\frac{1}{2}} b^{\frac{1}{2}} \left(a^{\frac{1}{6}} b^{\frac{-1}{6}} - 1\right)}$

$= \dfrac{ab}{a^{\frac{1}{2}} b^{\frac{1}{2}}}$

$= \dfrac{ab \times \sqrt{ab}}{\sqrt{ab} \times \sqrt{ab}}$

$= \dfrac{ab \times \sqrt{ab}}{ab}$

$= \sqrt{ab}$

Algebra - Algebraic Expressions

Simplify: $\;$ $2 \left(x^2 + \sqrt{x^4 - 1}\right) \left[\sqrt[3]{\left(x^2 + 1\right) \sqrt{1 + \dfrac{1}{x^2}}} + \sqrt[3]{\left(x^2 - 1\right) \sqrt{1 - \dfrac{1}{x^2}}}\right]^{-2}$

$2 \left(x^2 + \sqrt{x^4 - 1}\right) \left[\sqrt[3]{\left(x^2 + 1\right) \sqrt{1 + \dfrac{1}{x^2}}} + \sqrt[3]{\left(x^2 - 1\right) \sqrt{1 - \dfrac{1}{x^2}}}\right]^{-2}$ $\;\;\; \cdots \; (1)$

Consider the expression $\;\;$ $\sqrt[3]{\left(x^2 + 1\right) \sqrt{1 + \dfrac{1}{x^2}}}$

$= \left[\dfrac{\left(x^2 + 1\right) \left(x^2 + 1\right)^{\frac{1}{2}}}{x}\right]^{\frac{1}{3}}$

$= \left[\dfrac{\left(x^2 + 1\right)^{\frac{3}{2}}}{x}\right]^{\frac{1}{3}}$

$= x^{\frac{-1}{3}} \left(x^2 + 1\right)^{\frac{1}{2}}$ $\;\;\; \cdots \; (2)$

Consider the expression $\;\;$ $\sqrt[3]{\left(x^2 - 1\right) \sqrt{1 - \dfrac{1}{x^2}}}$

$= \left[\dfrac{\left(x^2 - 1\right) \left(x^2 - 1\right)^{\frac{1}{2}}}{x}\right]^{\frac{1}{3}}$

$= \left[\dfrac{\left(x^2 - 1\right)^{\frac{3}{2}}}{x}\right]^{\frac{1}{3}}$

$= x^{\frac{-1}{3}} \left(x^2 - 1\right)^{\frac{1}{2}}$ $\;\;\; \cdots \; (3)$

In view of expressions $(2)$ and $(3)$, the expression $\;\;$ $\left[\sqrt[3]{\left(x^2 + 1\right) \sqrt{1 + \dfrac{1}{x^2}}} + \sqrt[3]{\left(x^2 - 1\right) \sqrt{1 - \dfrac{1}{x^2}}}\right]^{-2}$ $\;\;$ becomes

$\left[x^{\frac{-1}{3}} \left(x^2 + 1\right)^{\frac{1}{2}} + x^{\frac{-1}{3}} \left(x^2 - 1\right)^{\frac{1}{2}}\right]^{-2}$

$= \left[x^{\frac{-1}{3}} \left(\sqrt{x^2 + 1} + \sqrt{x^2 - 1}\right)\right]^{-2}$

$= \left[\dfrac{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}{x^{\frac{1}{3}}}\right]^{-2}$

$= \left[\dfrac{x^{\frac{1}{3}}}{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}\right]^2$

$= \dfrac{x^{\frac{2}{3}}}{x^2 + 1 + x^2 - 1 + 2 \sqrt{\left(x^2 + 1\right) \left(x^2 - 1\right)}}$

$= \dfrac{x^{\frac{2}{3}}}{ 2 \left(x^2 + \sqrt{x^4 - 1}\right)}$ $\;\;\; \cdots \; (4)$

$\therefore \;$ In view of expression $(4)$, expression $(1)$ becomes

$2 \left(x^2 + \sqrt{x^4 - 1}\right) \times \dfrac{x^{\frac{2}{3}}}{2 \left(x^2 + \sqrt{x^4 - 1}\right)}$

$= x^{\frac{2}{3}} = \sqrt[3]{x^2}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\left[\left(\dfrac{\sqrt[3]{x^2 y^2} + x \sqrt[3]{x}}{x \sqrt[3]{y} + y \sqrt[3]{x}} - 1\right)^{-1} \left(1 + \sqrt[3]{\dfrac{x}{y}} + \sqrt[3]{\dfrac{x^2}{y^2}}\right)^{-1} + 1\right]^{\frac{1}{3}} \sqrt[3]{x - y}$

$\left[\left(\dfrac{\sqrt[3]{x^2 y^2} + x \sqrt[3]{x}}{x \sqrt[3]{y} + y \sqrt[3]{x}} - 1\right)^{-1} \left(1 + \sqrt[3]{\dfrac{x}{y}} + \sqrt[3]{\dfrac{x^2}{y^2}}\right)^{-1} + 1\right]^{\frac{1}{3}} \sqrt[3]{x - y}$ $\;\;\; \cdots \; (1)$

Consider the expression $\;\;$ $\left(\dfrac{\sqrt[3]{x^2 y^2} + x \sqrt[3]{x}}{x \sqrt[3]{y} + y \sqrt[3]{x}} - 1\right)^{-1}$

$= \left(\dfrac{x^{\frac{2}{3}} y^{\frac{2}{3}} + x^{\frac{4}{3}} - x y^{\frac{1}{3}} - y x^{\frac{1}{3}}}{x y^{\frac{1}{3}} + y x^{\frac{1}{3}}} \right)^{-1}$

$= \left[\dfrac{x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + y^{\frac{2}{3}}\right) - x^{\frac{1}{3}} y^{\frac{1}{3}} \left(x^{\frac{2}{3}} + y^{\frac{2}{3}}\right)}{x^{\frac{1}{3}} y^{\frac{1}{3}} \left(x^{\frac{2}{3}} + y^{\frac{2}{3}}\right)} \right]^{-1}$

$= \left[\dfrac{x^{\frac{1}{3}} \left(x^{\frac{1}{3}} - y^{\frac{1}{3}}\right)}{x^{\frac{1}{3}} y^{\frac{1}{3}}}\right]^{-1}$

$\dfrac{y^{\frac{1}{3}}}{x^{\frac{1}{3}} - y^{\frac{1}{3}}}$ $\;\;\; \cdots \; (2)$

Consider the expression $\;\;$ $\left(1 + \sqrt[3]{\dfrac{x}{y}} + \sqrt[3]{\dfrac{x^2}{y^2}}\right)^{-1}$

$= \left(1 + x^{\frac{1}{3}} y^{\frac{-1}{3}} + x^{\frac{2}{3}} y^{\frac{-2}{3}}\right)^{-1}$

$= \left[1 + x^{\frac{1}{3}} y^{\frac{-2}{3}} \left(x^{\frac{1}{3}} + y^{\frac{1}{3}}\right)\right]^{-1}$

$= \left[\dfrac{y^{\frac{2}{3}} + x^{\frac{1}{3}} \left(x^{\frac{1}{3}} + y^{\frac{1}{3}}\right)}{y^{\frac{2}{3}}}\right]^{-1}$

$= \dfrac{y^{\frac{2}{3}}}{y^{\frac{2}{3}} + x^{\frac{2}{3}} + x^{\frac{1}{3}} y^{\frac{1}{3}}}$ $\;\;\; \cdots \; (3)$

$\therefore \;$ In view of expressions $(2)$ and $(3)$, expression $\;$ $\left[\left(\dfrac{\sqrt[3]{x^2 y^2} + x \sqrt[3]{x}}{x \sqrt[3]{y} + y \sqrt[3]{x}} - 1\right)^{-1} \left(1 + \sqrt[3]{\dfrac{x}{y}} + \sqrt[3]{\dfrac{x^2}{y^2}}\right)^{-1} + 1\right]^{\frac{1}{3}}$ $\;$ becomes

$\left[\left(\dfrac{y^{\frac{1}{3}}}{x^{\frac{1}{3}} - y^{\frac{1}{3}}}\right) \left(\dfrac{y^{\frac{2}{3}}}{x^{\frac{2}{3}} + x^{\frac{1}{3}} y^{\frac{1}{3}} + y^{\frac{2}{3}}}\right) + 1\right]^{\frac{1}{3}}$

$= \left[\dfrac{y}{\left(x^{\frac{1}{3}}\right)^3 - \left(y^{\frac{1}{3}}\right)^3} + 1\right]^{\frac{1}{3}}$

$= \left[\dfrac{y}{x - y} + 1\right]^{\frac{1}{3}}$

$= \left[\dfrac{x}{x - y}\right]^{\frac{1}{3}}$ $\;\;\; \cdots \; (4)$

$\therefore \;$ In view of expression $(4)$, expression $(1)$ becomes

$\dfrac{\sqrt[3]{x}}{\sqrt[3]{x - y}} \times \sqrt[3]{x - y}$

$= \sqrt[3]{x}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\left[\dfrac{\left(\sqrt[4]{a} + \sqrt[4]{b}\right)^2 - \sqrt[4]{16ab}}{a - b} + \dfrac{1}{\sqrt{a} + \sqrt{b}} - \left(\dfrac{a - b}{2 \sqrt{b}}\right)^{-1}\right]^{-1}$

$\left[\dfrac{\left(\sqrt[4]{a} + \sqrt[4]{b}\right)^2 - \sqrt[4]{16ab}}{a - b} + \dfrac{1}{\sqrt{a} + \sqrt{b}} - \left(\dfrac{a - b}{2 \sqrt{b}}\right)^{-1}\right]^{-1}$

$= \left[\dfrac{\sqrt{a} + \sqrt{b} + 2 \sqrt[4]{ab} - 2 \sqrt[4]{ab}}{a - b} + \dfrac{\sqrt{a} - \sqrt{b}}{\left(\sqrt{a} + \sqrt{b}\right) \left(\sqrt{a} - \sqrt{b}\right)} - \dfrac{2 \sqrt{b}}{a - b} \right]^{-1}$

$= \left[\dfrac{\sqrt{a} + \sqrt{b}}{a - b} + \dfrac{\sqrt{a} - \sqrt{b}}{a - b} - \dfrac{2 \sqrt{b}}{a - b}\right]^{-1}$

$= \left[\dfrac{2 \sqrt{a} - 2 \sqrt{b}}{a - b}\right]^{-1}$

$= \dfrac{a - b}{2 \left(\sqrt{a} - \sqrt{b}\right)}$

$= \dfrac{\left(a - b\right) \left(\sqrt{a} + \sqrt{b}\right)}{2 \left(\sqrt{a} - \sqrt{b}\right) \left(\sqrt{a} + \sqrt{b}\right)}$

$= \dfrac{\left(a - b\right) \left(\sqrt{a} + \sqrt{b}\right)}{2 \left(a - b\right)}$

$= \dfrac{\sqrt{a} + \sqrt{b}}{2}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\dfrac{a + 10 \sqrt{a} + \sqrt{20} \left(\sqrt[4]{a^3} + 5 \sqrt[4]{a}\right) + 25}{\left(a - 25\right) \left(\sqrt[4]{a^3} - \sqrt{125}\right)^{-1} \left(\sqrt{a} + \sqrt[4]{25a} + 5\right)}$

$\dfrac{a + 10 \sqrt{a} + \sqrt{20} \left(\sqrt[4]{a^3} + 5 \sqrt[4]{a}\right) + 25}{\left(a - 25\right) \left(\sqrt[4]{a^3} - \sqrt{125}\right)^{-1} \left(\sqrt{a} + \sqrt[4]{25a} + 5\right)}$ $\;\;\; \cdots \; (1)$

Consider the expression

${\left(a - 25\right) \left(\sqrt[4]{a^3} - \sqrt{125}\right)^{-1} \left(\sqrt{a} + \sqrt[4]{25a} + 5\right)}$

$= \left(a - 25\right) \left[\left(a^{\frac{1}{4}}\right)^3 - \left(5^{\frac{1}{2}}\right)^3\right]^{-1} \left(a^{\frac{1}{2}} + 25^{\frac{1}{4}} a^{\frac{1}{4}} + 5\right)$

$= \left(a - 25\right) \left[\left(a^{\frac{1}{4}} - 5^{\frac{1}{2}}\right) \left(a^{\frac{1}{2}} + 5^{\frac{1}{2}} a^{\frac{1}{4}} + 5\right)\right]^{-1} \left(a^{\frac{1}{2}} + 25^{\frac{1}{4}} a^{\frac{1}{4}} + 5\right)$

$= \dfrac{\left(a - 25\right) \left(a^{\frac{1}{2}} + 25^{\frac{1}{4}} a^{\frac{1}{4}} + 5\right)}{\left(a^{\frac{1}{4}} - 5^{\frac{1}{2}}\right) \left(a^{\frac{1}{2}} + 5^{\frac{1}{2}} a^{\frac{1}{4}} + 5\right)}$

$= \dfrac{\left(a - 25\right) \left(a^{\frac{1}{2}} + 25^{\frac{1}{4}} a^{\frac{1}{4}} + 5\right)}{\left(a^{\frac{1}{4}} - 5^{\frac{1}{2}}\right) \left(a^{\frac{1}{2}} + 25^{\frac{1}{4}} a^{\frac{1}{4}} + 5\right)}$

$= \dfrac{a - 25}{a^{\frac{1}{4}} - 5^{\frac{1}{2}}}$

$= \dfrac{\left(a^{\frac{1}{2}} + 5\right) \left(a^{\frac{1}{2}} - 5\right)}{a^{\frac{1}{4}} - \sqrt{5}}$ $\;\;\; \cdots \; (2)$

Consider the expression

$a + 10 \sqrt{a} + \sqrt{20} \left(\sqrt[4]{a^3} + 5 \sqrt[4]{a}\right) + 25$

$= a + 10 a^{\frac{1}{2}} + 2 \sqrt{5} a^{\frac{3}{4}} + 10 \sqrt{5} a^{\frac{1}{4}} + 25$

$= \left(a + 10 a^{\frac{1}{2}} + 25\right) + 2 \sqrt{5} a^{\frac{1}{4}} \left(a^{\frac{1}{2}} + 5\right)$

$= \left[\left(a^{\frac{1}{2}}\right)^2 + 2 \times 5 \times a^{\frac{1}{2}} + 5^2\right] + 2 \sqrt{5} a^{\frac{1}{4}} \left(a^{\frac{1}{2}} + 5\right)$

$= \left(a^{\frac{1}{2}} + 5\right)^2 + 2 \sqrt{5} a^{\frac{1}{4}} \left(a^{\frac{1}{2}} + 5\right)$

$= \left(a^{\frac{1}{2}} + 5\right) \left(a^{\frac{1}{2}} + 5 + 2 \sqrt{5} a^{\frac{1}{4}}\right)$

$= \left(a^{\frac{1}{2}} + 5\right) \left[\left(a^{\frac{1}{4}}\right)^2 + 2 \times \sqrt{5} \times a^{\frac{1}{4}} + \left(\sqrt{5}\right)^2\right]$

$= \left(a^{\frac{1}{2}} + 5\right) \left(a^{\frac{1}{4}} + \sqrt{5}\right)^2$ $\;\;\; \cdots \; (3)$

In view of expressions $(2)$ and $(3)$, expression $(1)$ becomes

$\dfrac{\left(a^{\frac{1}{2}} + 5\right) \left(a^{\frac{1}{4}} + \sqrt{5}\right)^2}{\dfrac{\left(a^{\frac{1}{2}} + 5\right) \left(a^{\frac{1}{2}} - 5\right)}{a^{\frac{1}{4}} - \sqrt{5}}}$

$= \dfrac{\left(a^{\frac{1}{4}} + \sqrt{5}\right)^2 \left(a^{\frac{1}{4}} - \sqrt{5}\right)}{a^{\frac{1}{2}} - 5}$

$= \dfrac{\left(a^{\frac{1}{4}} + \sqrt{5}\right) \left(a^{\frac{1}{4}} + \sqrt{5}\right) \left(a^{\frac{1}{4}} - \sqrt{5}\right)}{a^{\frac{1}{2}} - 5}$

$= \dfrac{\left(a^{\frac{1}{4}} + \sqrt{5}\right) \left(a^{\frac{1}{2}} - 5\right)}{a^{\frac{1}{2}} - 5}$

$= \sqrt[4]{a} + \sqrt{5}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\dfrac{\sqrt{ab} \left(\sqrt{a} - \sqrt{b}\right)}{\left(\sqrt[4]{b} - \sqrt[4]{a}\right)^2 \sqrt[4]{b}} - \dfrac{\sqrt[4]{16 ab} \left(a + \sqrt[4]{a^3 b} + \sqrt{ab}\right)}{\sqrt[4]{a^3} - \sqrt[4]{b^3}}$

$\dfrac{\sqrt{ab} \left(\sqrt{a} - \sqrt{b}\right)}{\left(\sqrt[4]{b} - \sqrt[4]{a}\right)^2 \sqrt[4]{b}} - \dfrac{\sqrt[4]{16 ab} \left(a + \sqrt[4]{a^3 b} + \sqrt{ab}\right)}{\sqrt[4]{a^3} - \sqrt[4]{b^3}}$

$= \dfrac{a^{\frac{1}{2}} b^{\frac{1}{4}} \left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right)}{b^{\frac{1}{2}} + a^{\frac{1}{2}} - 2 b^{\frac{1}{4}} a^{\frac{1}{4}}} - \dfrac{2 a^{\frac{1}{4}} b^{\frac{1}{4}} \left(a + a^{\frac{3}{4}} b^{\frac{1}{4}} + a^{\frac{1}{2}} b^{\frac{1}{2}}\right)}{\left(a^{\frac{1}{4}}\right)^3 - \left(b^{\frac{1}{4}}\right)^3}$

$= \dfrac{a^{\frac{1}{2}} b^{\frac{1}{4}} \left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right)}{\left(a^{\frac{1}{4}} - b^{\frac{1}{4}}\right)^2} - \dfrac{2 a^{\frac{1}{4}} b^{\frac{1}{4}} a^{\frac{1}{2}} \left(a^{\frac{1}{2}} + a^{\frac{1}{4}} b^{\frac{1}{4}} + b^{\frac{1}{2}}\right)}{\left(a^{\frac{1}{4}} - b^{\frac{1}{4}}\right) \left(a^{\frac{1}{2}} + a^{\frac{1}{4}} b^{\frac{1}{4}} + b^{\frac{1}{2}}\right)}$

$= \dfrac{a^{\frac{1}{2}} b^{\frac{1}{4}} \left(a^{\frac{1}{4}} - b^{\frac{1}{4}}\right) \left(a^{\frac{1}{4}} + b^{\frac{1}{4}}\right)}{\left(a^{\frac{1}{4}} - b^{\frac{1}{4}}\right)^2} - \dfrac{2 a^{\frac{3}{4}} b^{\frac{1}{4}}}{a^{\frac{1}{4}} - b^{\frac{1}{4}}}$

$= \dfrac{a^{\frac{1}{2}} b^{\frac{1}{4}} \left(a^{\frac{1}{4}} + b^{\frac{1}{4}}\right)}{a^{\frac{1}{4}} - b^{\frac{1}{4}}} - \dfrac{2 a^{\frac{3}{4}} b^{\frac{1}{4}}}{a^{\frac{1}{4}} - b^{\frac{1}{4}}}$

$= \dfrac{a^{\frac{3}{4}} b^{\frac{1}{4}} + a^{\frac{1}{2}} b^{\frac{1}{2}} - 2 a^{\frac{3}{4}} b^{\frac{1}{4}}}{a^{\frac{1}{4}} - b^{\frac{1}{4}}}$

$= \dfrac{a^{\frac{1}{2}} b^{\frac{1}{2}} - a^{\frac{3}{4}} b^{\frac{1}{4}}}{a^{\frac{1}{4}} - b^{\frac{1}{4}}}$

$= \dfrac{a^{\frac{1}{2}} b^{\frac{1}{4}} \left(b^{\frac{1}{4}} - a^{\frac{1}{4}}\right)}{a^{\frac{1}{4}} - b^{\frac{1}{4}}}$

$= - a^{\frac{1}{2}} b^{\frac{1}{4}}$

Algebra - Algebraic Expressions

Simplify: $\;$ $- \left[\left(\dfrac{\sqrt{a} + \sqrt{b}}{\sqrt[4]{a} - \sqrt[4]{b}}\right)^{-1} - \dfrac{2 \sqrt[4]{ab}}{b^{\frac{3}{4}} - a^{\frac{1}{4}} b^{\frac{1}{2}} + a^{\frac{1}{2}} b^{\frac{1}{4}} - a^{\frac{3}{4}}}\right]^{-1} + \sqrt{2}^{\log_4 a}$

$- \left[\left(\dfrac{\sqrt{a} + \sqrt{b}}{\sqrt[4]{a} - \sqrt[4]{b}}\right)^{-1} - \dfrac{2 \sqrt[4]{ab}}{b^{\frac{3}{4}} - a^{\frac{1}{4}} b^{\frac{1}{2}} + a^{\frac{1}{2}} b^{\frac{1}{4}} - a^{\frac{3}{4}}}\right]^{-1} + \sqrt{2}^{\log_4 a}$

$= - \left[\dfrac{\sqrt[4]{a} - \sqrt[4]{b}}{\sqrt{a} + \sqrt{b}} - \dfrac{2 a^{\frac{1}{4}} b^{\frac{1}{4}}}{b^{\frac{1}{4}} \left(b^{\frac{1}{2}} + a^{\frac{1}{2}}\right) - a^{\frac{1}{4}} \left(b^{\frac{1}{2}} + a^{\frac{1}{2}}\right)}\right]^{-1} + \sqrt{2}^{\frac{\log_{\sqrt{2}} a}{\log_{\sqrt{2}} 4}}$

$= - \left[\dfrac{a^{\frac{1}{4}} - b^{\frac{1}{4}}}{a^{\frac{1}{2}} + b^{\frac{1}{2}}} - \dfrac{2 a^{\frac{1}{4}} b^{\frac{1}{4}}}{\left(b^{\frac{1}{4}} - a^{\frac{1}{4}}\right) \left(b^{\frac{1}{2}} + a^{\frac{1}{2}}\right)}\right]^{-1} + \sqrt{2}^{\left(\frac{\log_{\sqrt{2}} a}{\log_{\sqrt{2}} \sqrt{2}^4}\right)}$

$= - \left[\dfrac{- \left(b^{\frac{1}{4}} - a^{\frac{1}{4}}\right)^2 - 2 a^{\frac{1}{4}} b^{\frac{1}{4}}}{\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right) \left(b^{\frac{1}{4}} - a^{\frac{1}{4}}\right)}\right]^{-1} + \sqrt{2}^{\left(\frac{\log_{\sqrt{2}}a}{4 \log_{\sqrt{2}} \sqrt{2}}\right)}$

$= - \left[\dfrac{- \left(b^{\frac{1}{2}} + a^{\frac{1}{2}} - 2a^{\frac{1}{4}} b^{\frac{1}{4}}\right) - 2 a^{\frac{1}{4}} b^{\frac{1}{4}}}{\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right) \left(b^{\frac{1}{4}} - a^{\frac{1}{4}}\right)}\right]^{-1} + \sqrt{2}^{\left(\frac{1}{4} \log_{\sqrt{2}}a\right)}$

$= - \left[\dfrac{2 a^{\frac{1}{4}} b^{\frac{1}{4}} - a^{\frac{1}{2}} - b^{\frac{1}{2}} - 2 a^{\frac{1}{4}} b^{\frac{1}{4}}}{\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right) \left(b^{\frac{1}{4}} - a^{\frac{1}{4}}\right)}\right]^{-1} + \sqrt{2}^{\left(\log_{\sqrt{2}}a^{\frac{1}{4}}\right)}$

$= - \left[\dfrac{- \left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right)}{\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right) \left(b^{\frac{1}{4}} - a^{\frac{1}{4}}\right)}\right]^{-1} + a^{\frac{1}{4}}$

$= - \left[\dfrac{-1}{b^{\frac{1}{4}} - a^{\frac{1}{4}}}\right]^{-1} + a^{\frac{1}{4}}$

$= b^{\frac{1}{4}} - a^{\frac{1}{4}} + a^{\frac{1}{4}}$

$= b^{\frac{1}{4}}$

$= \sqrt[4]{b}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\left[\dfrac{3 - \sqrt{a}}{9 - a} + \dfrac{1}{3 - \sqrt{a}} - 6 \dfrac{a^2 + 162}{729 - a^3}\right]^{-1} + \dfrac{a \left(a + 9\right)}{54}$

$\left[\dfrac{3 - \sqrt{a}}{9 - a} + \dfrac{1}{3 - \sqrt{a}} - 6 \dfrac{a^2 + 162}{729 - a^3}\right]^{-1} + \dfrac{a \left(a + 9\right)}{54}$

$= \left[\dfrac{3 - \sqrt{a}}{\left(3\right)^2 - \left(\sqrt{a}\right)^2} + \dfrac{1}{3 - \sqrt{a}} - \dfrac{6 \left(a^2 + 162\right)}{\left(9\right)^3 - \left(a\right)^3}\right]^{-1} + \dfrac{a \left(a + 9\right)}{54}$

$= \left[\dfrac{3 - \sqrt{a}}{\left(3 + \sqrt{a}\right) \left(3 - \sqrt{a}\right)} + \dfrac{1}{3 - \sqrt{a}} - \dfrac{6 \left(a^2 + 162\right)}{\left(9 - a\right) \left(81 + 9a + a^2\right)}\right]^{-1} + \dfrac{a \left(a + 9\right)}{54}$

$= \left[\dfrac{1}{3 + \sqrt{a}} + \dfrac{1}{3 - \sqrt{a}} - \dfrac{6 \left(a^2 + 162\right)}{\left(9 - a\right) \left(81 + 9a + a^2\right)}\right]^{-1} + \dfrac{a \left(a + 9\right)}{54}$

$= \left[\dfrac{3 - \sqrt{a} + 3 + \sqrt{a}}{\left(3 + \sqrt{a}\right) \left(3 - \sqrt{a}\right)} - \dfrac{6 \left(a^2 + 162\right)}{\left(9 - a\right) \left(81 + 9a + a^2\right)}\right]^{-1} + \dfrac{a \left(a + 9\right)}{54}$

$= \left[\dfrac{6}{9 - a} - \dfrac{6 \left(a^2 + 162\right)}{\left(9 - a\right) \left(81 + 9a + a^2\right)}\right]^{-1} + \dfrac{a \left(a + 9\right)}{54}$

$= \left[\dfrac{6}{9 - a} \left(1 - \dfrac{a^2 + 162}{81 + 9a + a^2}\right)\right]^{-1} + \dfrac{a \left(a + 9\right)}{54}$

$= \left[\dfrac{6}{9 - a} \left(\dfrac{81 + 9a + a^2 - a^2 - 162}{81 + 9a + a^2}\right)\right]^{-1} + \dfrac{a \left(a + 9\right)}{54}$

$= \left[\dfrac{6 \left(9a - 81\right)}{\left(9 - a\right) \left(81 + 9a + a^2\right)}\right]^{-1} + \dfrac{a \left(a + 9\right)}{54}$

$= \left[\dfrac{6 \times 9 \left(a - 9\right)}{\left(9 - a\right) \left(81 + 9a + a^2\right)}\right]^{-1} + \dfrac{a \left(a + 9\right)}{54}$

$= \left[\dfrac{-54}{81 + 9a + a^2}\right]^{-1} + \dfrac{a^2 + 9a}{54}$

$= \dfrac{- \left(81 + 9a + a^2\right)}{54} + \dfrac{a^2 + 9a}{54}$

$= \dfrac{a^2 + 9a - a^2 - 9a - 81}{54}$

$= \dfrac{-81}{54}$

$= \dfrac{-3}{2}$

Algebraic Expressions

Simplify: $\;$ $\dfrac{a^2 + 10a + 25 + 2 \sqrt{5} \left(\sqrt{a^3} + 5 \sqrt{a}\right)}{\left(a^2 - 25\right) \left[\left(\sqrt{a^3} - \sqrt{125}\right) \left(a + \sqrt{5a} + 5\right)^{-1}\right]^{-1}}$

$\dfrac{a^2 + 10a + 25 + 2 \sqrt{5} \left(\sqrt{a^3} + 5 \sqrt{a}\right)}{\left(a^2 - 25\right) \left[\left(\sqrt{a^3} - \sqrt{125}\right) \left(a + \sqrt{5a} + 5\right)^{-1}\right]^{-1}}$

$= \dfrac{\left(a + 5\right)^2 + 2 \sqrt{5 a} \left(a + 5\right)}{\left(a + 5\right) \left(a - 5\right) \left[\dfrac{\left(\sqrt{a}\right)^3 - \left(\sqrt{5}\right)^3}{a + \sqrt{5a} + 5}\right]^{-1}}$

$= \dfrac{\left(a + 5\right) \left(a + 5 + 2 \sqrt{5a}\right)}{\dfrac{\left(a + 5\right) \left(a - 5\right) \left(a + \sqrt{5a} + 5\right)}{\left(\sqrt{a}\right)^3 - \left(\sqrt{5}\right)^3}}$

$= \dfrac{\left(a + 2 \sqrt{5a} + 5\right) \left[\left(\sqrt{a}\right)^3 - \left(\sqrt{5}\right)^3\right]}{\left(a - 5\right) \left(a + \sqrt{5a} + 5\right)}$

$= \dfrac{\left(\sqrt{a} + \sqrt{5}\right)^2 \left(\sqrt{a} - \sqrt{5}\right) \left(a + \sqrt{5a} + 5\right)}{\left[\left(\sqrt{a}\right)^2 - \left(\sqrt{5}\right)^2\right] \left(a + \sqrt{5a} + 5\right)}$

$= \dfrac{\left(\sqrt{a} + \sqrt{5}\right)^2 \left(\sqrt{a} - \sqrt{5}\right)}{\left(\sqrt{a} + \sqrt{5}\right) \left(\sqrt{a} - \sqrt{5}\right)}$

$= \sqrt{a} + \sqrt{5}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\dfrac{\left(\dfrac{\sqrt[4]{bx^3} + \sqrt[4]{a^2 bx}}{\sqrt{x} + \sqrt{a}} + \sqrt[4]{bx}\right)^2 + bx + 4}{x \left(\sqrt{b} + \sqrt{4 x^{-1}}\right)^2}$

$\dfrac{\left(\dfrac{\sqrt[4]{bx^3} + \sqrt[4]{a^2 bx}}{\sqrt{x} + \sqrt{a}} + \sqrt[4]{bx}\right)^2 + bx + 4}{x \left(\sqrt{b} + \sqrt{4 x^{-1}}\right)^2}$

$= \dfrac{\left(\dfrac{b^{\frac{1}{4}} x^{\frac{3}{4}} + a^{\frac{1}{2}} b^{\frac{1}{4}} x^{\frac{1}{4}}}{x^{\frac{1}{2}} + a^{\frac{1}{2}}} + b^{\frac{1}{4}} x^{\frac{1}{4}}\right)^2 + bx + 4}{x \left(\sqrt{b} + \dfrac{2}{\sqrt{x}}\right)^2}$

$= \dfrac{\left(\dfrac{b^{\frac{1}{4}} x^{\frac{3}{4}} + a^{\frac{1}{2}} b^{\frac{1}{4}} x^{\frac{1}{4}} + b^{\frac{1}{4}} x^{\frac{3}{4}} + a^{\frac{1}{2}} b^{\frac{1}{4}} x^{\frac{1}{4}}}{x^{\frac{1}{2}} + a^{\frac{1}{2}}}\right)^2 + bx + 4}{\left(\sqrt{bx} + 2\right)^2}$

$= \dfrac{\left(\dfrac{2 b^{\frac{1}{4}} x^{\frac{3}{4}} + 2 a^{\frac{1}{2}} b^{\frac{1}{4}} x^{\frac{1}{4}}}{x^{\frac{1}{2}} + a^{\frac{1}{2}}}\right)^2 + bx + 4}{\left(\sqrt{bx} + 2\right)^2}$

$= \dfrac{\left(\dfrac{2 b^{\frac{1}{4}} x^{\frac{1}{4}} \left(x^{\frac{1}{2}} + a^{\frac{1}{2}}\right)}{x^{\frac{1}{2}} + a^{\frac{1}{2}}}\right)^2 + bx + 4}{\left(\sqrt{bx} + 2\right)^2}$

$= \dfrac{4 b^{\frac{1}{2}} x^{\frac{1}{2}} + bx + 4}{bx + 4 + 4 b^{\frac{1}{2}} x^{\frac{1}{2}}}$

$= 1$

Algebra - Algebraic Expressions

Simplify: $\;$ $\left(2 - \dfrac{a}{4} - \dfrac{4}{a}\right) \left[\left(a - 4\right) \sqrt[3]{\left(a - 4\right)^{-3}} - \dfrac{\left(a^2 - 16\right)^{\frac{-1}{2}} \left(a - 4\right)^{\frac{-1}{2}}}{\left(a + 4\right)^{\frac{-3}{2}}}\right] \left(\dfrac{a + 4}{a - 4}\right)$

$\left(2 - \dfrac{a}{4} - \dfrac{4}{a}\right) \left[\left(a - 4\right) \sqrt[3]{\left(a - 4\right)^{-3}} - \dfrac{\left(a^2 - 16\right)^{\frac{-1}{2}} \left(a - 4\right)^{\frac{-1}{2}}}{\left(a + 4\right)^{\frac{-3}{2}}}\right] \left(\dfrac{a + 4}{a - 4}\right)$

$= \left(\dfrac{8a - a^2 - 16}{4a}\right) \left[\left(a - 4\right) \left(a - 4\right)^{-3 \times \frac{1}{3}} - \dfrac{\left(a + 4\right)^{\frac{-1}{2}} \left(a - 4\right)^{\frac{-1}{2}} \left(a - 4\right)^{\frac{-1}{2}}}{\left(a + 4\right)^{\frac{-3}{2}}} \right] \left(\dfrac{a + 4}{a - 4}\right)$

$= \left[\dfrac{- \left(a^2 - 8a + 16\right)}{4a}\right] \left[\dfrac{a - 4}{a - 4} - \dfrac{\left(a + 4\right)^{\frac{-1}{2} + \frac{3}{2}}}{a - 4}\right] \left(\dfrac{a + 4}{a - 4}\right)$

$= \dfrac{-\left(a - 4\right)^2}{4a} \left(1 - \dfrac{a + 4}{a - 4}\right) \left(\dfrac{a + 4}{a - 4}\right)$

$= \dfrac{-\left(a - 4\right)^2}{4a} \left(\dfrac{-8}{a - 4}\right) \left(\dfrac{a + 4}{a - 4}\right)$

$= \dfrac{2 \left(a + 4\right)}{a}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\left(\dfrac{\sqrt[4]{a^3} - b}{\sqrt[4]{a} - \sqrt[3]{b}} - 3 \sqrt[12]{a^3 b^4}\right)^{\frac{-1}{2}} \left(\dfrac{\sqrt[4]{a^3} + b}{\sqrt[4]{a} + \sqrt[3]{b}} - \sqrt[3]{b^2}\right)$, $\;$ $b > 0$, $\;$ $\sqrt[4]{a} > \sqrt[3]{b}$

$\left(\dfrac{\sqrt[4]{a^3} - b}{\sqrt[4]{a} - \sqrt[3]{b}} - 3 \sqrt[12]{a^3 b^4}\right)^{\frac{-1}{2}} \left(\dfrac{\sqrt[4]{a^3} + b}{\sqrt[4]{a} + \sqrt[3]{b}} - \sqrt[3]{b^2}\right)$

$= \left(\dfrac{a^{\frac{3}{4}} - b}{a^{\frac{1}{4}} - b^{\frac{1}{3}}} - 3 a^{\frac{1}{4}} b^{\frac{1}{3}}\right)^{\frac{-1}{2}} \left(\dfrac{a^{\frac{3}{4}} + b}{a^{\frac{1}{4}} + b^{\frac{1}{3}}} - b^{\frac{2}{3}}\right)$

$= \left(\dfrac{a^{\frac{3}{4}} - b - 3 a^{\frac{1}{2}} b^{\frac{1}{3}} + 3 a^{\frac{1}{4}} b^{\frac{2}{3}}}{a^{\frac{1}{4}} - b^{\frac{1}{3}}}\right)^{\frac{-1}{2}} \left(\dfrac{a^{\frac{3}{4}} + b - a^{\frac{1}{4}} b^{\frac{2}{3}} - b}{a^{\frac{1}{4}} + b^{\frac{1}{3}}}\right)$

$= \left[\dfrac{a^{\frac{1}{4}} - b^{\frac{1}{3}}}{\left(a^{\frac{1}{4}} - b^{\frac{1}{3}}\right)^3}\right]^{\frac{1}{2}} \left(\dfrac{a^{\frac{3}{4}} - a^{\frac{1}{4}} b^{\frac{2}{3}}}{a^{\frac{1}{4}} + b^{\frac{1}{3}}}\right)$

$= \left[\dfrac{1}{\left(a^{\frac{1}{4}} - b^{\frac{1}{3}}\right)^2}\right]^{\frac{1}{2}} \left[\dfrac{a^{\frac{1}{4}} \left(a^{\frac{1}{2}} - b^{\frac{2}{3}}\right)}{a^{\frac{1}{4}} + b^{\frac{1}{3}}}\right]$

$= \dfrac{a^{\frac{1}{4}} \left(a^{\frac{1}{2}} - b^{\frac{2}{3}}\right)}{\left(a^{\frac{1}{4}} - b^{\frac{1}{3}}\right) \left(a^{\frac{1}{4}} + b^{\frac{1}{3}}\right)}$

$= \dfrac{a^{\frac{1}{4}} \left(a^{\frac{1}{2}} - b^{\frac{2}{3}}\right)}{\left(a^{\frac{1}{4}}\right)^2 - \left(b^{\frac{1}{3}}\right)^2}$

$= \dfrac{a^{\frac{1}{4}} \left(a^{\frac{1}{2}} - b^{\frac{2}{3}}\right)}{a^{\frac{1}{2}} - b^{\frac{2}{3}}}$

$= a^{\frac{1}{4}}$ $\;\;\;$ $\left[\because \; \sqrt[4]{a} > \sqrt[3]{b} \implies \left(\sqrt[4]{a}\right)^2 > \left(\sqrt[3]{b}\right)^2 \implies a^{\frac{1}{2}} - b^{\frac{2}{3}} > 0 \right]$

Algebra - Algebraic Expressions

Simplify: $\;$ $\left[\dfrac{m - n}{m^{\frac{3}{4}} + m^{\frac{1}{2}} n^{\frac{1}{4}}} - \dfrac{m^{\frac{1}{2}} - n^{\frac{1}{2}}}{m^{\frac{1}{4}} + n^{\frac{1}{4}}}\right] \left[\dfrac{n}{m}\right]^{\frac{-1}{2}}$

$\left[\dfrac{m - n}{m^{\frac{3}{4}} + m^{\frac{1}{2}} n^{\frac{1}{4}}} - \dfrac{m^{\frac{1}{2}} - n^{\frac{1}{2}}}{m^{\frac{1}{4}} + n^{\frac{1}{4}}}\right] \left[\dfrac{n}{m}\right]^{\frac{-1}{2}}$

$= \left[\dfrac{m - n}{m^{\frac{1}{2}} \left(m^{\frac{1}{4}} + n^{\frac{1}{4}}\right)} - \dfrac{m^{\frac{1}{2}} - n^{\frac{1}{2}}}{m^{\frac{1}{4}} + n^{\frac{1}{4}}}\right] \times \dfrac{m^{\frac{1}{2}}}{n^{\frac{1}{2}}}$

$= \left[\dfrac{m - n - m + m^{\frac{1}{2}} n^{\frac{1}{2}}}{m^{\frac{1}{2}} \left(m^{\frac{1}{4}} + n^{\frac{1}{4}}\right)}\right] \times \dfrac{m^{\frac{1}{2}}}{n^{\frac{1}{2}}}$

$= \dfrac{n^{\frac{1}{2}} \left(m^{\frac{1}{2}} - n^{\frac{1}{2}}\right)}{m^{\frac{1}{2}} \left(m^{\frac{1}{4}} + n^{\frac{1}{4}}\right)} \times \dfrac{m^{\frac{1}{2}}}{n^{\frac{1}{2}}}$

$= \dfrac{m^{\frac{1}{2}} - n^{\frac{1}{2}}}{m^{\frac{1}{4}} + n^{\frac{1}{4}}}$

$= \dfrac{\left(m^{\frac{1}{2}} - n^{\frac{1}{2}}\right) \left(m^{\frac{1}{4}} - n^{\frac{1}{4}}\right)}{\left(m^{\frac{1}{4}} + n^{\frac{1}{4}}\right) \left(m^{\frac{1}{4}} - n^{\frac{1}{4}}\right)}$

$= \dfrac{\left(m^{\frac{1}{2}} - n^{\frac{1}{2}}\right) \left(m^{\frac{1}{4}} - n^{\frac{1}{4}}\right)}{m^{\frac{1}{2}} - n^{\frac{1}{2}}}$

$= m^{\frac{1}{4}} - n^{\frac{1}{4}}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\left[x \sqrt[3]{\dfrac{x - 1}{\left(x + 1\right)^2}} + \dfrac{x - 1}{\sqrt[3]{\left(x^2 - 1\right)^2}}\right]^{\frac{-3}{5}} : \left(x^2 - 1\right)^{\frac{4}{5}}$

$\left[x \sqrt[3]{\dfrac{x - 1}{\left(x + 1\right)^2}} + \dfrac{x - 1}{\sqrt[3]{\left(x^2 - 1\right)^2}}\right]^{\frac{-3}{5}} : \left(x^2 - 1\right)^{\frac{4}{5}}$

$= \left[\dfrac{x \left(x - 1\right)^{\frac{1}{3}}}{\left(x + 1\right)^{\frac{2}{3}}} + \dfrac{x - 1}{\left(x^2 - 1\right)^{\frac{2}{3}}}\right]^{\frac{-3}{5}} : \left(x + 1\right)^{\frac{4}{5}} \left(x - 1\right)^{\frac{4}{5}}$

$= \left[\dfrac{x \left(x - 1\right)^{\frac{1}{3}}}{\left(x + 1\right)^{\frac{2}{3}}} + \dfrac{x - 1}{\left(x + 1\right)^{\frac{2}{3}} \left(x - 1\right)^{\frac{2}{3}}}\right]^{\frac{-3}{5}} : \left(x + 1\right)^{\frac{4}{5}} \left(x - 1\right)^{\frac{4}{5}}$

$= \left[\dfrac{x \left(x - 1\right)^{\frac{1}{3}} + \left(x - 1\right)^{\frac{1}{3}}}{\left(x + 1\right)^{\frac{2}{3}}}\right]^{\frac{-3}{5}} : \left(x + 1\right)^{\frac{4}{5}} \left(x - 1\right)^{\frac{4}{5}}$

$= \left[\dfrac{\left(x + 1\right)^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{1}{3}} \left(x + 1\right)}\right]^{\frac{3}{5}} \times \dfrac{1}{\left(x + 1\right)^{\frac{4}{5}} \left(x - 1\right)^{\frac{4}{5}}}$

$= \dfrac{1}{\left[\left(x - 1\right)^{\frac{1}{3}}\right]^{\frac{3}{5}} \left[\left(x + 1\right)^{\frac{1}{3}}\right]^{\frac{3}{5}}} \times \dfrac{1}{\left(x + 1\right)^{\frac{4}{5}} \left(x - 1\right)^{\frac{4}{5}}}$

$= \dfrac{1}{\left(x - 1\right)^{\frac{1}{5}} \left(x + 1\right)^{\frac{1}{5}} \left(x + 1\right)^{\frac{4}{5}} \left(x - 1\right)^{\frac{4}{5}}}$

$= \dfrac{1}{\left(x - 1\right) \left(x + 1\right)}$

$= \dfrac{1}{x^2 - 1}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\left(1 - 2 \sqrt[3]{\dfrac{b}{a}}\right) \left(\dfrac{a^{\frac{4}{3}} - 8 a^{\frac{1}{3}}b}{a^{\frac{2}{3}} + 2 \sqrt[3]{ab} + 4 b^{\frac{2}{3}}}\right)^{-1} \sqrt[3]{\dfrac{1}{a^{-2}}}$

$\left(1 - 2 \sqrt[3]{\dfrac{b}{a}}\right) \left(\dfrac{a^{\frac{4}{3}} - 8 a^{\frac{1}{3}}b}{a^{\frac{2}{3}} + 2 \sqrt[3]{ab} + 4 b^{\frac{2}{3}}}\right)^{-1} \sqrt[3]{\dfrac{1}{a^{-2}}}$

$= \left(1 - \dfrac{2 b^{\frac{1}{3}}}{a^{\frac{1}{3}}}\right) \left(\dfrac{a^{\frac{1}{3}} \left(a - 8b\right)}{a^{\frac{2}{3}} + 2 a^{\frac{1}{3}} b^{\frac{1}{3}} + 4b^{\frac{2}{3}}}\right)^{-1} \times a^{\frac{2}{3}}$

$= \left(\dfrac{a^{\frac{1}{3}} - 2b^{\frac{1}{3}}}{a^{\frac{1}{3}}}\right) \times \left(\dfrac{a^{\frac{2}{3}} + 2 a^{\frac{1}{3}} b^{\frac{1}{3}} + 4 b^{\frac{2}{3}}}{a^{\frac{1}{3}} \left(a - 8b\right)}\right) \times a^{\frac{2}{3}}$

$= \dfrac{\left(a^{\frac{1}{3}} - 2 b^{\frac{1}{3}}\right) \left(a^{\frac{2}{3}} + 2a^{\frac{1}{3}} b^{\frac{1}{3}} + 4 b^{\frac{2}{3}}\right) \times a^{\frac{2}{3}}}{a^{\frac{2}{3}} \left[\left(a^{\frac{1}{3}}\right)^3 - \left(2b^{\frac{1}{3}}\right)^3\right]}$

$= \dfrac{\left(a^{\frac{1}{3}} - 2 b^{\frac{1}{3}}\right) \left(a^{\frac{2}{3}} + 2a^{\frac{1}{3}} b^{\frac{1}{3}} + 4 b^{\frac{2}{3}}\right) }{\left(a^{\frac{1}{3}} - 2 b^{\frac{1}{3}}\right) \left(a^{\frac{2}{3}} + 2a^{\frac{1}{3}} b^{\frac{1}{3}} + 4 b^{\frac{2}{3}}\right)}$

$= 1$

Algebra - Algebraic Expressions

Simplify: $\;$ $\dfrac{\left(\sqrt[8]{a} + \sqrt[8]{b}\right)^2 + \left(\sqrt[8]{a} - \sqrt[8]{b}\right)^2}{a - \sqrt{ab}} : \dfrac{\left(\sqrt[4]{a} + \sqrt[8]{ab} + \sqrt[4]{b}\right) \left(\sqrt[4]{a} - \sqrt[8]{ab} + \sqrt[4]{b}\right)}{\sqrt[4]{a^3 b} - b}$

Given expression: $\dfrac{\left(\sqrt[8]{a} + \sqrt[8]{b}\right)^2 + \left(\sqrt[8]{a} - \sqrt[8]{b}\right)^2}{a - \sqrt{ab}} : \dfrac{\left(\sqrt[4]{a} + \sqrt[8]{ab} + \sqrt[4]{b}\right) \left(\sqrt[4]{a} - \sqrt[8]{ab} + \sqrt[4]{b}\right)}{\sqrt[4]{a^3 b} - b}$ $\;\;\; \cdots \; (1)$

Consider the expression $\;\;\;$ $\dfrac{\left(\sqrt[8]{a} + \sqrt[8]{b}\right)^2 + \left(\sqrt[8]{a} - \sqrt[8]{b}\right)^2}{a - \sqrt{ab}}$

$= \dfrac{a^{\frac{1}{4}} + b^{\frac{1}{4}} + 2 a^{\frac{1}{8}} b^{\frac{1}{8}} + a^{\frac{1}{4}} + b^{\frac{1}{4}} - 2 a^{\frac{1}{8}} b^{\frac{1}{8}}}{\sqrt{a} \left(\sqrt{a} - \sqrt{b}\right)}$

$= \dfrac{2 \left(a^{\frac{1}{4}} + b^{\frac{1}{4}}\right)}{\sqrt{a} \left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right)}$

$= \dfrac{2 \left(a^{\frac{1}{4}} + b^{\frac{1}{4}}\right)}{\sqrt{a} \left[\left(a^{\frac{1}{4}}\right)^2 - \left(b^{\frac{1}{4}}\right)^2\right]}$

$= \dfrac{2 \left(a^{\frac{1}{4}} + b^{\frac{1}{4}}\right)}{\sqrt{a} \left(a^{\frac{1}{4}} + b^{\frac{1}{4}}\right) \left(a^{\frac{1}{4}} - b^{\frac{1}{4}}\right)}$

$= \dfrac{2}{\sqrt{a} \left(\sqrt[4]{a} - \sqrt[4]{b}\right)}$ $\;\;\; \cdots \; (2)$

Consider the expression $\;\;\;$ $\dfrac{\left(\sqrt[4]{a} + \sqrt[8]{ab} + \sqrt[4]{b}\right) \left(\sqrt[4]{a} - \sqrt[8]{ab} + \sqrt[4]{b}\right)}{\sqrt[4]{a^3 b} - b}$

$= \dfrac{\left(\sqrt[4]{a} + \sqrt[4]{b}\right)^2 - \left(\sqrt[8]{ab}\right)^2}{a^{\frac{3}{4}} b^{\frac{1}{4}} - b}$

$= \dfrac{a^{\frac{1}{2}} + b^{\frac{1}{2}} + 2 a^{\frac{1}{4}} b^{\frac{1}{4}} - a^{\frac{1}{4}} b^{\frac{1}{4}}}{b^{\frac{1}{4}} \left(a^{\frac{3}{4}} - b^{\frac{3}{4}}\right)}$

$= \dfrac{a^{\frac{1}{2}} + b^{\frac{1}{2}} + a^{\frac{1}{4}} b^{\frac{1}{4}}}{b^{\frac{1}{4}} \left[\left(a^{\frac{1}{4}}\right)^3 - \left(b^{\frac{1}{4}}\right)^3\right]}$

$= \dfrac{a^{\frac{1}{2}} + b^{\frac{1}{2}} + a^{\frac{1}{4}} b^{\frac{1}{4}}}{b^{\frac{1}{4}} \left(a^{\frac{1}{4}} - b^{\frac{1}{4}}\right) \left(a^{\frac{1}{2}} + a^{\frac{1}{4}} b^{\frac{1}{4}} + b^{\frac{1}{2}}\right)}$

$= \dfrac{1}{\sqrt[4]{b} \left(\sqrt[4]{a} - \sqrt[4]{b}\right)}$ $\;\;\; \cdots \; (3)$

In view of expressions $(2)$ and $(3)$, expression $(1)$ becomes

$\dfrac{2}{\sqrt{a} \left(\sqrt[4]{a} - \sqrt[4]{b}\right)} : \dfrac{1}{\sqrt[4]{b} \left(\sqrt[4]{a} - \sqrt[4]{b}\right)}$

$= \dfrac{2 \sqrt[4]{b}}{\sqrt{a}}$

$= 2 \sqrt[4]{\dfrac{b}{a^2}}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\dfrac{\left(a - b\right)^3 \left(\sqrt{a} + \sqrt{b}\right)^{-3} + 2 a \sqrt{a} + b \sqrt{b}}{a \sqrt{a} + b \sqrt{b}} - \dfrac{3 \left(\sqrt{ab} - b\right)}{a - b}$

$\dfrac{\left(a - b\right)^3 \left(\sqrt{a} + \sqrt{b}\right)^{-3} + 2 a \sqrt{a} + b \sqrt{b}}{a \sqrt{a} + b \sqrt{b}} - \dfrac{3 \left(\sqrt{ab} - b\right)}{a - b}$

$= \dfrac{\dfrac{\left(a - b\right)^3}{\left(\sqrt{a} + \sqrt{b}\right)^3} + 2 a \sqrt{a} + b \sqrt{b}}{a \sqrt{a} + b \sqrt{b}} - \dfrac{3 \sqrt{b} \left(\sqrt{a} - \sqrt{b}\right)}{\left(\sqrt{a}\right)^2 - \left(\sqrt{b}\right)^2}$

$= \dfrac{\dfrac{\left[\left(\sqrt{a}\right)^2 - \left(\sqrt{b}\right)^2\right]^3}{\left(\sqrt{a} + \sqrt{b}\right)^3} + 2 a \sqrt{a} + b \sqrt{b}}{a \sqrt{a} + b \sqrt{b}} - \dfrac{3 \sqrt{b} \left(\sqrt{a} - \sqrt{b}\right)}{\left(\sqrt{a} + \sqrt{b}\right) \left(\sqrt{a} - \sqrt{b}\right)}$

$= \dfrac{\dfrac{\left(\sqrt{a} + \sqrt{b}\right)^3 \left(\sqrt{a} - \sqrt{b}\right)^3}{\left(\sqrt{a} + \sqrt{b}\right)^3} + 2 a \sqrt{a} + b \sqrt{b}}{a \sqrt{a} + b \sqrt{b}} - \dfrac{3 \sqrt{b}}{\sqrt{a} + \sqrt{b}}$

$= \dfrac{\left(\sqrt{a} - \sqrt{b}\right)^3 + 2 a \sqrt{a} + b \sqrt{b}}{a \sqrt{a} + b \sqrt{b}} - \dfrac{3 \sqrt{b} }{\sqrt{a} + \sqrt{b}}$

$= \dfrac{a \sqrt{a} - 3 a \sqrt{b} + 3 b \sqrt{a} - b \sqrt{b} + 2 a \sqrt{a} + b \sqrt{b}}{a \sqrt{a} + b \sqrt{b}} - \dfrac{3 \sqrt{b}}{\sqrt{a} + \sqrt{b}}$

$= \dfrac{3 a \sqrt{a} - 3 a \sqrt{b} + 3 b \sqrt{a}}{a \sqrt{a} + b \sqrt{b}} - \dfrac{3 \sqrt{b}}{\sqrt{a} + \sqrt{b}}$

$= \dfrac{3a^2 - 3 a \sqrt{ab} + 3ab + 3 a \sqrt{ab} - 3ab + 3b \sqrt{ab} - 3a \sqrt{ab} - 3b^2}{\left(a \sqrt{a} + b \sqrt{b}\right) \left(\sqrt{a} + \sqrt{b}\right)}$

$= \dfrac{3a^2 + 3b \sqrt{ab} - 3a \sqrt{ab} - 3b^2}{\left(a \sqrt{a} + b \sqrt{b}\right) \left(\sqrt{a} + \sqrt{b}\right)}$

$= \dfrac{3 \left[\left(a^2 - b^2\right) + \sqrt{ab} \left(b - a\right)\right]}{\left(a \sqrt{a} + b \sqrt{b}\right) \left(\sqrt{a} + \sqrt{b}\right)}$

$= \dfrac{3 \left[\left(a + b\right) \left(a - b\right) + \sqrt{ab} \left(b - a\right)\right]}{\left(a \sqrt{a} + b \sqrt{b}\right) \left(\sqrt{a} + \sqrt{b}\right)}$

$= \dfrac{3 \left(a - b\right) \left(a + b - \sqrt{ab}\right) }{\left(a \sqrt{a} + b \sqrt{b}\right) \left(\sqrt{a} + \sqrt{b}\right)}$

$= \dfrac{3 \left(a - b\right) \left(a + b - \sqrt{a}\right) \left(a \sqrt{a} - b \sqrt{b}\right)}{\left(\sqrt{a} + \sqrt{b}\right) \left(a \sqrt{a} + b \sqrt{b}\right) \left(a \sqrt{a} - b \sqrt{b}\right)}$

$= \dfrac{3 \left(a - b\right) \left(a^2 \sqrt{a} + ab \sqrt{a} - a^2 \sqrt{b} - ab \sqrt{b} - b^2 \sqrt{b} + b^2 \sqrt{a}\right)}{\left(\sqrt{a} + \sqrt{b}\right) \left(a^3 - b^3\right)}$

$= \dfrac{3 \left(a - b\right) \left[a^2 \left(\sqrt{a} - \sqrt{b}\right) + ab \left(\sqrt{a} - \sqrt{b}\right) + b^2 \left(\sqrt{a} - \sqrt{b}\right)\right] }{\left(\sqrt{a} + \sqrt{b}\right) \left(a - b\right) \left(a^2 + ab + b^2\right)}$

$= \dfrac{3 \left(\sqrt{a} - \sqrt{b}\right) \left(a^2 + ab + b^2\right)}{\left(\sqrt{a} + \sqrt{b}\right) \left(a^2 + ab + b^2\right)}$

$= \dfrac{3 \left(\sqrt{a} - \sqrt{b}\right)}{\sqrt{a} + \sqrt{b}}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) : \left[\left(\dfrac{\sqrt[3]{y}}{y \sqrt{x}}\right)^{\frac{3}{2}} + \left(\dfrac{x^{\frac{-1}{2}}}{\sqrt[8]{y^3}}\right)^2\right]$

$\left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) : \left[\left(\dfrac{\sqrt[3]{y}}{y \sqrt{x}}\right)^{\frac{3}{2}} + \left(\dfrac{x^{\frac{-1}{2}}}{\sqrt[8]{y^3}}\right)^2\right]$

$= \left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) : \left[\dfrac{y^{\frac{1}{3} \times \frac{3}{2}}}{y^{\frac{3}{2}} x^{\frac{1}{2} \times \frac{3}{2}}} + \dfrac{x^{\frac{-1}{2} \times 2}}{y^{\frac{3}{8} \times 2}}\right]$

$= \left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) : \left[\dfrac{y^{\frac{1}{2} - \frac{3}{2}}}{x^{\frac{3}{4}}} + \dfrac{x^{-1}}{y^{\frac{3}{4}}} \right]$

$= \left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) : \left[\dfrac{y^{-1}}{x^{\frac{3}{4}}} + \dfrac{x^{-1}}{y^{\frac{3}{4}}} \right]$

$= \left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) : \left[\dfrac{y^{-1 + \frac{3}{4}} + x^{-1 + \frac{3}{4}}}{x^{\frac{3}{4}} y^{\frac{3}{4}}}\right]$

$= \left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) : \left[\dfrac{y^{\frac{-1}{4}} + x^{\frac{-1}{4}}}{x^{\frac{3}{4}} y^{\frac{3}{4}}}\right]$

$= \left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) : \left[\dfrac{\dfrac{1}{y^{\frac{1}{4}}} + \dfrac{1}{x^{\frac{1}{4}}}}{x^{\frac{3}{4}} y^{\frac{3}{4}}}\right]$

$= \left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) : \left[\dfrac{x^{\frac{1}{4}} + y^{\frac{1}{4}}}{x^{\frac{1}{4}} y^{\frac{1}{4}} \times x^{\frac{3}{4}} y^{\frac{3}{4}}}\right]$

$= \left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right) \times \dfrac{xy}{\left(x^{\frac{1}{4}} + y^{\frac{1}{4}}\right)}$

$= xy$

Algebra - Algebraic Expressions

Simplify: $\;$ $\dfrac{1 + a \sqrt[3]{a} + a + \sqrt[3]{a^2}}{1 - \sqrt[3]{a}} + \dfrac{1}{\sqrt[3]{a^{-2}}}$

$\dfrac{1 + a \sqrt[3]{a} + a + \sqrt[3]{a^2}}{1 - \sqrt[3]{a}} + \dfrac{1}{\sqrt[3]{a^{-2}}}$

$= \dfrac{1 + a^{\frac{4}{3}} + a + a^{\frac{2}{3}}}{1 - a^{\frac{1}{3}}} + a^{\frac{2}{3}}$

$= \dfrac{1 + a^{\frac{4}{3}} + a + a^{\frac{2}{3}} + a^{\frac{2}{3}} - a}{1 - a^{\frac{1}{3}}}$

$= \dfrac{1 + a^{\frac{4}{3}} + 2 a^{\frac{2}{3}}}{1 - a^{\frac{1}{3}}}$

$= \dfrac{\left(1 + a^{\frac{2}{3}}\right)^2}{1 - a^{\frac{1}{3}}}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\left(\dfrac{1 - x^{-2}}{x^{\frac{1}{2}} - x^{\frac{-1}{2}}} - \dfrac{2}{x^2 : \sqrt{x}} + \dfrac{x^{-2} - x}{x^{\frac{1}{2}} - x^{\frac{-1}{2}}}\right) \left(1 + \dfrac{2}{x^2}\right)^{-1}$

$\left(\dfrac{1 - x^{-2}}{x^{\frac{1}{2}} - x^{\frac{-1}{2}}} - \dfrac{2}{x^2 : \sqrt{x}} + \dfrac{x^{-2} - x}{x^{\frac{1}{2}} - x^{\frac{-1}{2}}}\right) \left(1 + \dfrac{2}{x^2}\right)^{-1}$

$= \left(\dfrac{1 - \dfrac{1}{x^2}}{\sqrt{x} - \dfrac{1}{\sqrt{x}}} - \dfrac{2 \sqrt{x}}{x^2} + \dfrac{\dfrac{1}{x^2} - x}{\sqrt{x} - \dfrac{1}{\sqrt{x}}}\right) \left(\dfrac{x^2 + 2}{x^2}\right)^{-1}$

$= \left(\dfrac{\dfrac{x^2 - 1}{x^2}}{\dfrac{x - 1}{\sqrt{x}}} - \dfrac{2 \sqrt{x}}{x^2} + \dfrac{\dfrac{1 - x^3}{x^2}}{\dfrac{x - 1}{\sqrt{x}}}\right) \left(\dfrac{x^2}{x^2 + 2}\right)$

$= \left(\dfrac{\left(x + 1\right) \left(x - 1\right) \sqrt{x}}{x^2 \left(x - 1\right)} - \dfrac{2 \sqrt{x}}{x^2} + \dfrac{\sqrt{x} \left(1 - x\right) \left(1 + x + x^2\right)}{x^2 \left(x - 1\right)}\right) \left(\dfrac{x^2}{x^2 + 2}\right)$

$= \left(\sqrt{x} \left(x + 1\right) - 2 \sqrt{x} - \sqrt{x} \left(1 + x + x^2\right)\right) \times \dfrac{1}{x^2 + 2}$

$= \dfrac{x \sqrt{x} + \sqrt{x} - 2 \sqrt{x} - \sqrt{x} - x \sqrt{x} - x^2 \sqrt{x}}{x^2 + 2}$

$= \dfrac{- 2 \sqrt{x} - x^2 \sqrt{x}}{x^2 + 2}$

$= \dfrac{- \sqrt{x} \left(x^2 + 2\right)}{x^2 + 2}$

$= - \sqrt{x}$

Algebra - Algebraic Expressions

Simplify: $\;$ $a^2 \left(1 - a^2\right)^{\frac{-1}{2}} : \left\{\dfrac{1}{1 + \left[a \left(1 - a^2\right)^{\frac{-1}{2}}\right]^2} \times \dfrac{\left(1 - a^2\right)^{\frac{1}{2}} + a^2 \left(1 - a^2\right)^{\frac{-1}{2}}}{1 - a^2} \right\}$

$a^2 \left(1 - a^2\right)^{\frac{-1}{2}} : \left\{\dfrac{1}{1 + \left[a \left(1 - a^2\right)^{\frac{-1}{2}}\right]^2} \times \dfrac{\left(1 - a^2\right)^{\frac{1}{2}} + a^2 \left(1 - a^2\right)^{\frac{-1}{2}}}{1 - a^2} \right\}$

$= \dfrac{a^2}{\sqrt{1 - a^2}} : \left\{\dfrac{1}{1 + \dfrac{a^2}{1 - a^2}} \times \dfrac{\sqrt{1 - a^2} + \dfrac{a^2}{\sqrt{1 - a^2}}}{1 - a^2} \right\}$

$= \dfrac{a^2}{\sqrt{1 - a^2}} : \left\{\dfrac{1 - a^2}{1 - a^2 + a^2} \times \dfrac{1 - a^2 + a^2}{\left(1 - a^2\right) \sqrt{1 - a^2}} \right\}$

$= \dfrac{a^2}{\sqrt{1 - a^2}} : \dfrac{1}{\sqrt{1 - a^2}}$

$= \dfrac{a^2}{\sqrt{1 - a^2}} \times \sqrt{1 - a^2}$

$= a^2$

Algebra - Algebraic Expressions

Simplify: $\;$ $\dfrac{1}{2} \left[\left(\sqrt{a^3 b^{-3}} - \sqrt{b^3 a^{-3}}\right) : \left(\dfrac{a^2 + b^2}{ab} + 1\right)\right] \times \dfrac{2 \left(a - b\right)^{-1}}{\left(ab\right)^{\frac{-1}{2}}}$ $\;\;$ for $\;$ $a > 0$, $\;$ $b > 0$, $\;$ $a \neq b$

$\dfrac{1}{2} \left[\left(\sqrt{a^3 b^{-3}} - \sqrt{b^3 a^{-3}}\right) : \left(\dfrac{a^2 + b^2}{ab} + 1\right)\right] \times \dfrac{2 \left(a - b\right)^{-1}}{\left(ab\right)^{\frac{-1}{2}}}$

$= \left[\left(\sqrt{\dfrac{a^3}{b^3}} - \sqrt{\dfrac{b^3}{a^3}}\right) : \dfrac{a^2 + b^2 + ab}{ab}\right] \times \dfrac{\sqrt{ab}}{\left(a - b\right)}$

$= \dfrac{a^3 - b^3}{\sqrt{a^3 b^3}} \times \dfrac{ab}{a^2 + b^2 + ab} \times \dfrac{\sqrt{ab}}{a - b}$

$= \dfrac{\left(a - b\right) \left(a^2 + ab + b^2\right)}{ab \sqrt{ab}} \times \dfrac{ab}{a^2 + b^2 + ab} \times \dfrac{\sqrt{ab}}{\left(a - b\right)}$

$= 1$ $\;\;$ for $\;$ $a > 0$, $\;$ $b > 0$, $\;$ $a \neq b$

Algebra - Algebraic Expressions

Simplify: $\;$ $\dfrac{2x^{\frac{-1}{3}}}{x^{\frac{2}{3}} - 3 x^{\frac{-1}{3}}} - \dfrac{x^{\frac{2}{3}}}{x^{\frac{5}{3}} - x^{\frac{2}{3}}} - \dfrac{x + 1}{x^2 - 4x + 3}$

$\dfrac{2x^{\frac{-1}{3}}}{x^{\frac{2}{3}} - 3 x^{\frac{-1}{3}}} - \dfrac{x^{\frac{2}{3}}}{x^{\frac{5}{3}} - x^{\frac{2}{3}}} - \dfrac{x + 1}{x^2 - 4x + 3}$

$= \dfrac{\dfrac{2}{x^{\frac{1}{3}}}}{x^{\frac{2}{3}} - \dfrac{3}{x^{\frac{1}{3}}}} - \dfrac{x^{\frac{2}{3}}}{x^{\frac{2}{3}} \left(x - 1\right)} - \dfrac{x + 1}{x^2 - 3x - x + 3}$

$= \dfrac{2}{x - 3} - \dfrac{1}{x - 1} - \dfrac{x + 1}{\left(x - 3\right) \left(x - 1\right)}$

$= \dfrac{2 x - 2 - x + 3 - x - 1}{\left(x - 3\right) \left(x - 1\right)}$

$= \dfrac{2x -2x + 3 - 3}{\left(x - 3\right) \left(x - 1\right)}$

$= 0$

Algebra - Algebraic Expressions

Simplify: $\;$ $\left(\dfrac{2a + b^{\frac{1}{2}} a^{\frac{1}{2}}}{3a}\right)^{-1} \left(\dfrac{a^{\frac{3}{2}} - b^{\frac{3}{2}}}{a - a^{\frac{1}{2}} b^{\frac{1}{2}}} - \dfrac{a - b}{\sqrt{a} + \sqrt{b}} \right)$

Given expression $\;\;$ $\left(\dfrac{2a + b^{\frac{1}{2}} a^{\frac{1}{2}}}{3a}\right)^{-1} \left(\dfrac{a^{\frac{3}{2}} - b^{\frac{3}{2}}}{a - a^{\frac{1}{2}} b^{\frac{1}{2}}} - \dfrac{a - b}{\sqrt{a} + \sqrt{b}} \right)$ $\;\;\; \cdots \; (1)$

Consider the expression $\;\;$ $\left(\dfrac{2a + b^{\frac{1}{2}} a^{\frac{1}{2}}}{3a}\right)^{-1}$

$= \dfrac{3a}{2a + b^{\frac{1}{2}} a^{\frac{1}{2}}}$

$= \dfrac{3 \sqrt{a} \times \sqrt{a}}{2a + \sqrt{a} \sqrt{b}}$

$= \dfrac{3 \sqrt{a} \times \sqrt{a}}{\sqrt{a} \left(2 \sqrt{a} + \sqrt{b}\right)}$

$= \dfrac{3 \sqrt{a}}{2 \sqrt{a} + \sqrt{b}}$ $\;\;\; \cdots \; (2)$

Consider the expression $\;\;$ $\dfrac{a^{\frac{3}{2}} - b^{\frac{3}{2}}}{a - a^{\frac{1}{2}} b^{\frac{1}{2}}} - \dfrac{a - b}{\sqrt{a} + \sqrt{b}}$

$= \dfrac{\left(a^{\frac{1}{2}}\right)^3 - \left(b^{\frac{1}{2}}\right)^3}{a - \sqrt{a} \sqrt{b}} - \dfrac{\left(a - b\right) \left(\sqrt{a} - \sqrt{b}\right)}{\left(\sqrt{a} + \sqrt{b}\right) \left(\sqrt{a} - \sqrt{b}\right)}$

$= \dfrac{\left(\sqrt{a} - \sqrt{b}\right) \left(a + \sqrt{ab} + b\right)}{\sqrt{a} \left(\sqrt{a} - \sqrt{b}\right)} - \dfrac{\left(a - b\right) \left(\sqrt{a} - \sqrt{b}\right)}{a - b}$

$= \dfrac{a + \sqrt{ab} + b}{\sqrt{a}} - \left(\sqrt{a} - \sqrt{b}\right)$

$= \dfrac{a + \sqrt{ab} + b - a + \sqrt{ab}}{\sqrt{a}}$

$= \dfrac{2 \sqrt{ab} + b}{\sqrt{a}}$

$= \dfrac{\sqrt{b} \left(2 \sqrt{a} + \sqrt{b}\right)}{\sqrt{a}}$ $\;\;\; \cdots \; (3)$

In view of expressions $(2)$ and $(3)$, expression $(1)$ becomes

$\dfrac{3 \sqrt{a}}{2 \sqrt{a} + \sqrt{b}} \times \dfrac{\sqrt{b} \left(2 \sqrt{a} + \sqrt{b}\right)}{\sqrt{a}}$

$= 3 \sqrt{b}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\dfrac{a^{\frac{4}{3}} - 8 a^{\frac{1}{3}} b}{a^{\frac{2}{3}} + 2\sqrt[3]{ab} + 4 b^{\frac{2}{3}}} : \left(1 - 2 \sqrt[3]{\dfrac{b}{a}}\right)$

$\dfrac{a^{\frac{4}{3}} - 8 a^{\frac{1}{3}} b}{a^{\frac{2}{3}} + 2\sqrt[3]{ab} + 4 b^{\frac{2}{3}}} : \left(1 - 2 \sqrt[3]{\dfrac{b}{a}}\right)$

$= \dfrac{a^{\frac{1}{3}} \left(a - 8b\right)}{a^{\frac{2}{3}} + 2 a^{\frac{1}{3}} b^{\frac{1}{3}} + 4 b^{\frac{2}{3}}} \times \dfrac{1}{1 - 2 b^{\frac{1}{3}} a^{\frac{-1}{3}}}$

$= \dfrac{a^{\frac{1}{3}} \left(a - 8b\right)}{a^{\frac{2}{3}} + 2 a^{\frac{1}{3}} b^{\frac{1}{3}} + 4 b^{\frac{2}{3}} - 2 a^{\frac{1}{3}} b^{\frac{1}{3}} - 4b^{\frac{2}{3}} - 8 a^{\frac{-1}{3}} b}$

$= \dfrac{a^{\frac{1}{3}} \left(a - 8b\right)}{a^{\frac{2}{3}} - 8 a^{\frac{-1}{3}} b}$

$= \dfrac{a^{\frac{1}{3}} \left(a - 8b\right)}{a^{\frac{-1}{3}} \left(a - 8b\right)}$

$= \dfrac{a^{\frac{1}{3}}}{a^{\frac{-1}{3}}}$

$= a^{\frac{2}{3}}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\left[\left(\sqrt{\dfrac{a}{b}} - \sqrt{\dfrac{b}{a}}\right) : \left(\sqrt{\dfrac{a}{b}} + \sqrt{\dfrac{b}{a}} - 2\right)\right] : \left(1 + \sqrt{\dfrac{b}{a}}\right)$, $\;\;$ $a > 0, \; b> 0$

The given expression

$\left[\left(\sqrt{\dfrac{a}{b}} - \sqrt{\dfrac{b}{a}}\right) : \left(\sqrt{\dfrac{a}{b}} + \sqrt{\dfrac{b}{a}} - 2\right)\right] : \left(1 + \sqrt{\dfrac{b}{a}}\right)$ $\;\;\; \cdots \; (1)$

Consider $\;\;\;$ $\left(\sqrt{\dfrac{a}{b}} - \sqrt{\dfrac{b}{a}}\right) : \left(\sqrt{\dfrac{a}{b}} + \sqrt{\dfrac{b}{a}} - 2\right)$

$= \dfrac{a - b}{\sqrt{ab}} : \dfrac{a + b - 2 \sqrt{ab}}{\sqrt{ab}}$

$= \dfrac{a - b}{\left(\sqrt{a} - \sqrt{b}\right)^2}$

$= \dfrac{\left(\sqrt{a} + \sqrt{b}\right) \left(\sqrt{a} - \sqrt{b}\right)}{\left(\sqrt{a} - \sqrt{b}\right)^2}$

$= \dfrac{\sqrt{a} + \sqrt{b}}{\sqrt{a} - \sqrt{b}}$ $\;\;\; \cdots \; (2)$

$\therefore \;$ In view of $(2)$, expression $(1)$ becomes

$\dfrac{\sqrt{a} + \sqrt{b}}{\sqrt{a} - \sqrt{b}} : 1 + \sqrt{\dfrac{b}{a}}$

$= \dfrac{\sqrt{a} + \sqrt{b}}{\sqrt{a} - \sqrt{b}} \times \dfrac{\sqrt{a}}{\sqrt{a} + \sqrt{b}}$

$= \dfrac{\sqrt{a}}{\sqrt{a} - \sqrt{b}}$

Algebra - Algebraic Expressions

Simplify:
$\;$ $\left(\dfrac{2}{x^2 - a^2}\right)^{-1} \left\{\left[\dfrac{x \left(x^2 - a^2\right)^{\frac{-1}{2}} + 1}{a \left(x - a\right)^{\frac{-1}{2}} + \left(x - a\right)^{\frac{1}{2}}} : \left(\dfrac{x - \left(x^2 - a^2\right)^{\frac{1}{2}}}{a^2 \left(x + a\right)^{\frac{1}{2}}}\right)^{-1} \right] + \left(x^2 + ax\right)^{-1} \right\}$

The given expression

$\left(\dfrac{2}{x^2 - a^2}\right)^{-1} \left\{\left[\dfrac{x \left(x^2 - a^2\right)^{\frac{-1}{2}} + 1}{a \left(x - a\right)^{\frac{-1}{2}} + \left(x - a\right)^{\frac{1}{2}}} : \left(\dfrac{x - \left(x^2 - a^2\right)^{\frac{1}{2}}}{a^2 \left(x + a\right)^{\frac{1}{2}}}\right)^{-1} \right] + \left(x^2 + ax\right)^{-1} \right\}$ $\;\;\; \cdots \; (1)$

Consider $\;\;\;$ $\dfrac{x \left(x^2 - a^2\right)^{\frac{-1}{2}} + 1}{a \left(x - a\right)^{\frac{-1}{2}} + \left(x - a\right)^{\frac{1}{2}}} : \left(\dfrac{x - \left(x^2 - a^2\right)^{\frac{1}{2}}}{a^2 \left(x + a\right)^{\frac{1}{2}}}\right)^{-1}$

$= \dfrac{\dfrac{x}{\sqrt{x^2 - a^2}} + 1}{\dfrac{a}{\sqrt{x - a}} + \sqrt{x - a}} : \dfrac{a^2 \sqrt{x + a}}{x - \sqrt{x^2 - a^2}}$

$= \dfrac{\left(x + \sqrt{x^2 - a^2}\right) \sqrt{x - a}}{\left(a + x - a\right) \sqrt{x^2 - a^2}} : \dfrac{a^2 \sqrt{x + a}}{x - \sqrt{x^2 - a^2}}$

$= \dfrac{\left(x + \sqrt{x^2 - a^2}\right) \sqrt{x - a}}{x \sqrt{x + a} \sqrt{x - a}} \times \dfrac{\left(x - \sqrt{x^2 - a^2}\right)}{a^2 \sqrt{x + a}}$

$= \dfrac{x^2 - x^2 + a^2}{a^2 x \left(x + a\right)}$

$= \dfrac{a^2}{a^2 x \left(x + a\right)}$

$= \dfrac{1}{x \left(x + a\right)}$ $\;\;\; \cdots \; (2)$

In view of expression $(2)$, the expression

$\left[\dfrac{x \left(x^2 - a^2\right)^{\frac{-1}{2}} + 1}{a \left(x - a\right)^{\frac{-1}{2}} + \left(x - a\right)^{\frac{1}{2}}} : \left(\dfrac{x - \left(x^2 - a^2\right)^{\frac{1}{2}}}{a^2 \left(x + a\right)^{\frac{1}{2}}}\right)^{-1} \right] + \left(x^2 + ax\right)^{-1}$

becomes

$= \dfrac{1}{x \left(x + a\right)} + \left(x^2 + ax\right)^{-1}$

$= \dfrac{1}{x \left(x + a\right)} + \dfrac{1}{x^2 + ax}$

$= \dfrac{2}{x \left(x + a\right)}$ $\;\;\; \cdots \; (3)$

$\therefore \;$ In view of expression $(3)$, expression $(1)$ becomes

$\left(\dfrac{2}{x^2 - a^2}\right)^{-1} \times \dfrac{2}{x \left(x + a\right)}$

$= \dfrac{\left(x + a\right) \left(x - a\right)}{2} \times \dfrac{2}{x \left(x + a\right)}$

$= \dfrac{x - a}{x}$

Algebra - Algebraic Expressions

Simplify: $\;$ $\dfrac{\left(\sqrt{a^3} + \sqrt{b^3}\right)}{\sqrt[3]{a^2 \left(a - b\right)^2}} \times \dfrac{a^{\frac{2}{3}} \left(\sqrt{a^3} - \sqrt{b^3}\right)}{\left(a - b\right)^{\frac{1}{3}}}$

$\dfrac{\left(\sqrt{a^3} + \sqrt{b^3}\right)}{\sqrt[3]{a^2 \left(a - b\right)^2}} \times \dfrac{a^{\frac{2}{3}} \left(\sqrt{a^3} - \sqrt{b^3}\right)}{\left(a - b\right)^{\frac{1}{3}}}$

$= \dfrac{\left(a^3 - b^3\right) a^{\frac{2}{3}}}{a^{\frac{2}{3}} \left(a - b\right)^{\frac{2}{3}} \left(a - b\right)^{\frac{1}{3}}}$

$= \dfrac{\left(a - b\right) \left(a^2 + ab + b^2\right)}{a - b}$

$= a^2 + ab + b^2$

Algebra - Algebraic Expressions

Simplify: $\;$ $\left(\dfrac{x^{\frac{1}{2}} - y^{\frac{1}{2}}}{x y^{\frac{1}{2}} + y x^{\frac{1}{2}}} + \dfrac{x^{\frac{1}{2}} + y^{\frac{1}{2}}}{x y^{\frac{1}{2}} - y x^{\frac{1}{2}}}\right) \dfrac{x^{\frac{3}{2}} y^{\frac{1}{2}}}{x + y} - \dfrac{2y}{x - y}$

$\left(\dfrac{x^{\frac{1}{2}} - y^{\frac{1}{2}}}{x y^{\frac{1}{2}} + y x^{\frac{1}{2}}} + \dfrac{x^{\frac{1}{2}} + y^{\frac{1}{2}}}{x y^{\frac{1}{2}} - y x^{\frac{1}{2}}}\right) \dfrac{x^{\frac{3}{2}} y^{\frac{1}{2}}}{x + y} - \dfrac{2y}{x - y}$ $\;\;\; \cdots \; (1)$

Consider $\; \;$ $\left(\dfrac{x^{\frac{1}{2}} - y^{\frac{1}{2}}}{x y^{\frac{1}{2}} + y x^{\frac{1}{2}}} + \dfrac{x^{\frac{1}{2}} + y^{\frac{1}{2}}}{x y^{\frac{1}{2}} - y x^{\frac{1}{2}}}\right)$

$= \dfrac{x^{\frac{3}{2}} y^{\frac{1}{2}} - xy - xy + x^{\frac{1}{2}} y^{\frac{3}{2}} + x^{\frac{3}{2}} y^{\frac{1}{2}} + xy + xy + x^{\frac{1}{2}} y^{\frac{3}{2}}}{\left(x y^{\frac{1}{2}} + y x^{\frac{1}{2}}\right) \left(x y^{\frac{1}{2}} - y x^{\frac{1}{2}}\right)}$

$= \dfrac{2 x^{\frac{3}{2}} y^{\frac{1}{2}} + 2 x^{\frac{1}{2}} y^{\frac{3}{2}}}{x^2 y - y^2 x}$

$= \dfrac{2 x^{\frac{1}{2}} y^{\frac{1}{2}} \left(x + y\right)}{xy \left(x - y\right)}$

$= \dfrac{2 x^{\frac{-1}{2}} y^{\frac{-1}{2}} \left(x + y\right)}{x - y}$ $\;\;\; \cdots \; (2)$

In view of $(2)$, expression $(1)$ becomes

$\left[\dfrac{2 x^{\frac{-1}{2}} y^{\frac{-1}{2}} \left(x + y\right)}{\left(x - y\right)}\right] \times \dfrac{x^{\frac{3}{2}} y^{\frac{1}{2}}}{\left(x + y\right)} - \dfrac{2y}{x - y}$

$= \dfrac{2x}{x - y} - \dfrac{2y}{x - y}$

$= \dfrac{2 \left(x - y\right)}{x - y}$

$= 2$

Algebra - Algebraic Expressions

Simplify: $\;$ $\dfrac{2}{a} - \left\{\left[\dfrac{a + 1}{a^3 - 1} - \dfrac{1}{a^2 + a + 1} - \dfrac{2}{1 - a} \right] : \dfrac{a^3 + a^2 + 2a}{a^3 - 1} \right\}$

$\dfrac{2}{a} - \left\{\left[\dfrac{a + 1}{a^3 - 1} - \dfrac{1}{a^2 + a + 1} - \dfrac{2}{1 - a} \right] : \dfrac{a^3 + a^2 + 2a}{a^3 - 1} \right\}$ $\;\;\; \cdots \; (1)$

Consider the expression

$\left[\dfrac{a + 1}{a^3 - 1} - \dfrac{1}{a^2 + a + 1} - \dfrac{2}{1 - a} \right] : \dfrac{a^3 + a^2 + 2a}{a^3 - 1}$

$= \left[\dfrac{a + 1}{\left(a - 1\right) \left(a^2 + a + 1\right)} - \dfrac{1}{a^2 + a + 1} - \dfrac{2}{1 - a}\right] : \dfrac{a \left(a^2 + a + 2\right)}{\left(a - 1\right) \left(a^2 + a + 1\right)}$

$= \left[\dfrac{a + 1 - a + 1}{\left(a - 1\right) \left(a^2 + a + 1\right)} + \dfrac{2}{a - 1}\right] : \dfrac{a \left(a^2 + a + 2\right)}{\left(a - 1\right) \left(a^2 + a + 1\right)}$

$= \left[\dfrac{2}{\left(a - 1\right) \left(a^2 + a + 1\right)} + \dfrac{2}{a - 1}\right] : \dfrac{a \left(a^2 + a + 2\right)}{\left(a - 1\right) \left(a^2 + a + 1\right)}$

$= \dfrac{2 + 2 a^2 + 2a + 2}{\left(a - 1\right) \left(a^2 + a + 1\right)} : \dfrac{a \left(a^2 + a + 2\right)}{\left(a - 1\right) \left(a^2 + a + 1\right)}$

$= \dfrac{2a^2 + 2a + 4}{\left(a - 1\right) \left(a^2 + a + 1\right)} \times \dfrac{\left(a - 1\right) \left(a^2 + a + 1\right)}{a \left(a^2 + a + 2\right)}$

$= \dfrac{2 \left(a^2 + a + 2\right)}{\left(a - 1\right) \left(a^2 + a + 1\right)} \times \dfrac{\left(a - 1\right) \left(a^2 + a + 1\right)}{a \left(a^2 + a + 2\right)}$

$= \dfrac{2}{a}$ $\;\;\; \cdots \; (2)$

In view of $(2)$, expression $(1)$ becomes

$\dfrac{2}{a} - \dfrac{2}{a}$

$= 0$

Algebra - Algebraic Expressions

Simplify: $\;$ $\left\{\left(1 - a^2\right) : \left[\left(\dfrac{1 - a^{\frac{3}{2}}}{1 - a^{\frac{1}{2}}} + a^{\frac{1}{2}}\right) \left(\dfrac{1 + a^{\frac{3}{2}}}{1 + a^{\frac{1}{2}}} - a^{\frac{1}{2}}\right) \right] \right\} + 1$

$\left\{\left(1 - a^2\right) : \left[\left(\dfrac{1 - a^{\frac{3}{2}}}{1 - a^{\frac{1}{2}}} + a^{\frac{1}{2}}\right) \left(\dfrac{1 + a^{\frac{3}{2}}}{1 + a^{\frac{1}{2}}} - a^{\frac{1}{2}}\right) \right] \right\} + 1$

$= \left\{\left(1 - a^2\right) : \left[\left(\dfrac{1 - a^{\frac{3}{2}} + a^{\frac{1}{2}} - a}{1 - a^{\frac{1}{2}}}\right) \left(\dfrac{1 + a^{\frac{3}{2}} - a^{\frac{1}{2}} - a}{1 + a^{\frac{1}{2}}}\right) \right] \right\} + 1$

$= \left\{\left(1 - a^2\right) : \left[\left(\dfrac{\left(1 - a\right) + a^{\frac{1}{2}} \left(1 - a\right)}{1 - a^{\frac{1}{2}}}\right) \left(\dfrac{\left(1 - a\right) - a^{\frac{1}{2}} \left(1 - a\right)}{1 + a^{\frac{1}{2}}}\right)\right] \right\} + 1$

$= \left\{\left(1 - a^2\right) : \left[\dfrac{\left(1 + a^{\frac{1}{2}}\right) \left(1 - a\right)}{\left(1 - a^{\frac{1}{2}}\right)} \times \dfrac{\left(1 - a\right) \left(1 - a^{\frac{1}{2}}\right)}{\left(1 + a^{\frac{1}{2}}\right)}\right] \right\} + 1$

$= \left\{\left(1 - a^2\right) : \left(1 - a\right)^2 \right\} + 1$

$= \dfrac{1 - a^2}{\left(1 - a\right)^2} + 1$

$= \dfrac{1 - a^2 + \left(1 - a\right)^2}{\left(1 - a\right)^2}$

$= \dfrac{1 - a^2 + 1 + a^2 - 2a}{\left(1 - a\right)^2}$

$= \dfrac{2 - 2a}{\left(1 - a\right)^2}$

$= \dfrac{2 \left(1 - a\right)}{\left(1 - a\right)^2}$

$= \dfrac{2}{1 - a}$