Chapter A.10, Problem 24AYU

To determine

To find: Simplify the expression ${\left(\sqrt[3]{3}\sqrt{10}\right)}^4$.

Answer to Problem 24AYU

The value of ${\left(\sqrt[3]{3}\sqrt{10}\right)}^4=300\sqrt[3]{3}$.

Explanation of Solution

Given:

It is asked to find the simplest form of the given expression ${\left(\sqrt[3]{3}\sqrt{10}\right)}^4$.

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.

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

${\left(\sqrt[3]{3}\sqrt{10}\right)}^4=\left(\sqrt[3]{3}\sqrt{10}\right)\left(\sqrt[3]{3}\sqrt{10}\right)\left(\sqrt[3]{3}\sqrt{10}\right)\left(\sqrt[3]{3}\sqrt{10}\right)$

$=\sqrt[3]{3}\times \sqrt[3]{3}\times \sqrt[3]{3}\times \sqrt[3]{3}\times \sqrt{10}\times \sqrt{10}\times \sqrt{10}\times \sqrt{10}$

$={\left(3\right)}^{\frac{1}{3}}\times {\left(3\right)}^{\frac{1}{3}}\times {\left(3\right)}^{\frac{1}{3}}\times {\left(3\right)}^{\frac{1}{3}}\times {\left(10\right)}^{\frac{1}{2}}\times {\left(10\right)}^{\frac{1}{2}}\times {\left(10\right)}^{\frac{1}{2}}\times {\left(10\right)}^{\frac{1}{2}}$

$={\left(3\right)}^{\frac{1}{3}}\times {\left(3\right)}^{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}}\times {\left(10\right)}^{\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}}={\left(3\right)}^{\frac{1}{3}}\times 3\times {\left(10\right)}^2=300\sqrt[3]{3}$