Chapter A.10, Problem 23AYU

To determine

To find: Simplify the expression ${\left(\sqrt{5}\sqrt[3]{9}\right)}^2$.

Answer to Problem 23AYU

The value of ${\left(\sqrt{5}\sqrt[3]{9}\right)}^2=5\sqrt[3]{3}$.

Explanation of Solution

Given:

It is asked to find the simplest form of the given expression ${\left(\sqrt{5}\sqrt[3]{9}\right)}^2$.

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.

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

${\left(\sqrt{5}\sqrt[3]{9}\right)}^2=(\sqrt{5}\sqrt[3]{9)}\left(\sqrt{5}\sqrt[3]{9}\right)=\sqrt{5}\times \sqrt{5}\times \sqrt[3]{9}\times \sqrt[3]{9}=5\times {\left(9\right)}^{\frac{1}{3}}\times {\left(9\right)}^{\frac{1}{3}}$

$5\times {\left(9\right)}^{\frac{1}{3}}\times {\left(9\right)}^{\frac{1}{3}}=5\times {\left(3^2\right)}^{\frac{2}{3}}=5\times {\left(3\right)}^{\frac{1}{3}}=5\sqrt[3]{3}$