Chapter A.10, Problem 64AYU

To determine

To find: Simplify the expression ${\left(25\right)}^{\frac{3}{2}}$.

Answer to Problem 64AYU

The simplest form of ${\left(25\right)}^{\frac{3}{2}}$ is 125.

Explanation of Solution

Given:

It is asked to simplify the expression ${\left(25\right)}^{\frac{3}{2}}$.

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 ${\left(25\right)}^{\frac{3}{2}}$.

${\left(25\right)}^{\frac{3}{2}}=\sqrt{{25}^3}={\left(\sqrt{25}\right)}^3$

${\left(\sqrt{25}\right)}^3={\left(5\right)}^3=125$