Chapter A.10, Problem 61AYU

To determine

To find: Simplify the expression ${\left(-27\right)}^{\frac{1}{3}}$.

Answer to Problem 61AYU

The simplest form of ${\left(-27\right)}^{\frac{1}{3}}$ is $-3$.

Explanation of Solution

Given:

It is asked to simplify the expression ${\left(-27\right)}^{\frac{1}{3}}$.

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 ${\left(-27\right)}^{\frac{1}{3}}$.

${\left(-27\right)}^{\frac{1}{3}}={\left({\left(-3\right)}^3\right)}^{\frac{1}{3}}=-3$