Chapter A.10, Problem 12AYU

To determine

To find: Simplify the expression $\sqrt[3]{54}$.

Answer to Problem 12AYU

The value of $\sqrt[3]{54}=3\sqrt[3]{2}$.

Explanation of Solution

Given:

It is asked to find the simplest form of the given expression $\sqrt[3]{54}$.

Calculation:

If $n\ge 2$ is an integer and $a$ is a real number, then

$\sqrt[n]{a^n}=a$ if $n\ge 3$ is odd.

$\sqrt[n]{a^n}=\left|a\right|$ if $n\ge 2$ is even.

And

$\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$

$\sqrt[3]{54}=\sqrt[3]{2\times 27}=\sqrt[3]{3^3\times 2}$

$\sqrt[3]{3^3\times 2}=\sqrt[3]{{\left(3\right)}^3}\times \sqrt[3]{2}=3\sqrt[3]{2}\\$