Chapter A.10, Problem 60AYU

To determine

To find: Simplify the expression $4^{\frac{3}{2}}$.

Answer to Problem 60AYU

The simplest form of $4^{\frac{3}{2}}$ is 8.

Explanation of Solution

Given:

It is asked to simplify the expression $4^{\frac{3}{2}}$.

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 $4^{\frac{3}{2}}$.

$4^{\frac{3}{2}}=\sqrt{4^3}={\left(\sqrt{4}\right)}^3$

${\left(\sqrt{4}\right)}^3={\left(\sqrt{2^2}\right)}^3=2^3=8$