Chapter A.10, Problem 75AYU

To determine

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

Answer to Problem 75AYU

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

Explanation of Solution

Given:

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

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

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