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

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

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

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

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

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

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

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