Chapter A.10, Problem 76AYU

To determine

To find: Simplify the expression $\frac{{\left(xy\right)}^{\frac{1}{4}}{\left(x^2{y}^2\right)}^{\frac{1}{2}}}{{(x^{2}y)}^{\frac{3}{4}}}$.

Answer to Problem 76AYU

The simplest form of $\frac{{\left(xy\right)}^{\frac{1}{4}}{\left(x^2{y}^2\right)}^{\frac{1}{2}}}{{(x^{2}y)}^{\frac{3}{4}}}$ is $\frac{\sqrt{y}}{\sqrt[4]{x}}$.

Explanation of Solution

Given:

It is asked to simplify the expression $\frac{{\left(xy\right)}^{\frac{1}{4}}{\left(x^2{y}^2\right)}^{\frac{1}{2}}}{{(x^{2}y)}^{\frac{3}{4}}}$.

Calculation:

If $a$ is a real number and $m$ and $n$ are integers containing no common factors, with $n\ge 2$, then $a^{\frac{m}{n}}=\sqrt[n]{a^m}={\left(\sqrt[n]{a}\right)}^m$ provided that $\sqrt[n]{a}$ exists.

Consider $\frac{{\left(xy\right)}^{\frac{1}{4}}{\left(x^2{y}^2\right)}^{\frac{1}{2}}}{{(x^{2}y)}^{\frac{3}{4}}}$.

$\frac{{\left(xy\right)}^{\frac{1}{4}}{\left(x^2{y}^2\right)}^{\frac{1}{2}}}{{(x^{2}y)}^{\frac{3}{4}}}=\frac{\left(x^{\frac{1}{4}}{y}^{\frac{1}{4}}\right)\left(x^{\frac{2}{2}}{y}^{\frac{2}{2}}\right)}{x^{\frac{6}{4}}{y}^{\frac{3}{4}}}=\left(x^{\frac{1}{4}}{y}^{\frac{1}{4}}\right)\left(x^{\frac{2}{2}}{y}^{\frac{2}{2}}\right)\left(x^{-\frac{6}{4}}{y}^{-\frac{3}{4}}\right)=x^{\frac{1}{4}+1-\frac{6}{4}}\times y^{\frac{1}{4}+1-\frac{3}{4}}=x^{\frac{1+4-6}{4}}\times y^{\frac{1+4-3}{4}}=x^{-\frac{1}{4}}{y}^{\frac{1}{2}}=\frac{\sqrt{y}}{\sqrt[4]{x}}$