Chapter A.10, Problem 14AYU

To determine

To find: Simplify the expression $\sqrt[4]{48x^5}$.

Answer to Problem 14AYU

The value of $\sqrt[4]{48x^5}=2x\sqrt[4]{3x}$.

Explanation of Solution

Given:

It is asked to find the simplest form of the given expression $\sqrt[4]{48x^5}$.

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

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

$\sqrt[4]{48x^5}=\sqrt[4]{16\times 3\times x^4\times x}=\sqrt[4]{{\left(2\right)}^4\times x^4\times 3x}$

$\sqrt[4]{{\left(2\right)}^4\times x^4\times 3x}=\sqrt[4]{{\left(2\right)}^4}\times \sqrt[4]{x^4}\times \sqrt[4]{3x}=\left|2\right|x\sqrt[4]{3x}=2x\sqrt[4]{3x}$