Chapter A.10, Problem 74AYU

To determine

To find: Simplify the expression ${(x^4{y}^8)}^{\frac{3}{4}}$.

Answer to Problem 74AYU

The simplest form of ${(x^4{y}^8)}^{\frac{3}{4}}$ is $x^3{y}^6$.

Explanation of Solution

Given:

It is asked to simplify the expression ${(x^4{y}^8)}^{\frac{3}{4}}$.

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 ${(x^4{y}^8)}^{\frac{3}{4}}$.

${(x^4{y}^8)}^{\frac{3}{4}}={(x^4)}^{\frac{3}{4}}\times {\left(y^8\right)}^{\frac{3}{4}}={\left(\sqrt[4]{x^4}\right)}^3\times {\left(\sqrt[4]{y^8}\right)}^3=x^3\times y^6=x^3{y}^6$