Algebra - Algebraic Expressions

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

$\dfrac{\left(a^{\frac{1}{m}} - a^{\frac{1}{n}}\right)^2 + 4 \; a^{\frac{m + n}{mn}}}{\left(a^{\frac{2}{m}} - a^{\frac{2}{n}}\right) \left(\sqrt[m]{a^{m + 1}} + \sqrt[n]{a^{n + 1}}\right)}$

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

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

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

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

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

Algebra - Algebraic Expressions

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

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

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

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

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

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

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

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

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

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

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

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

Algebra - Algebraic Expressions

Simplify: $\;$ $\left[\dfrac{a^3 - 8}{a^2 - 5a + 6} - \dfrac{\left(a + 1\right)^2 + 3}{a - 3} + \dfrac{a^2 + a}{\sqrt[4]{a}}\right] : \dfrac{\sqrt{ab}}{\sqrt[4]{a^{-1} b^2}}$

$\left[\dfrac{a^3 - 8}{a^2 - 5a + 6} - \dfrac{\left(a + 1\right)^2 + 3}{a - 3} + \dfrac{a^2 + a}{\sqrt[4]{a}}\right] : \dfrac{\sqrt{ab}}{\sqrt[4]{a^{-1} b^2}}$

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

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

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

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

$= \dfrac{0 + a^{\frac{3}{4}} \left(a + 1\right)}{a^{\frac{3}{4}}}$

$= a + 1$

Algebra - Algebraic Expressions

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

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

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

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

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

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

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

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

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

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

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

Algebra - Algebraic Expressions

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

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

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

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

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

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

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

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

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

Algebra - Algebraic Expressions

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

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

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

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

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

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

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

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

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

Algebra - Algebraic Expressions

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

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

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

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

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

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

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

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

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

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

$= 1$