Chapter A.10, Problem 25AYU

To determine

To find: Simplify the expression $\left(3\sqrt{6}\right)\left(2\sqrt{2}\right)$.

Answer to Problem 25AYU

The value of $\left(3\sqrt{6}\right)\left(2\sqrt{2}\right)=12\sqrt{3}$.

Explanation of Solution

Given:

It is asked to find the simplest form of the given expression $\left(3\sqrt{6}\right)\left(2\sqrt{2}\right)$.

Calculation:

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

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

$\left(3\sqrt{6}\right)\left(2\sqrt{2}\right)=3\times 2\times \sqrt{6}\times \sqrt{2}=6\times {\left(6\right)}^{\frac{1}{2}}\times {\left(2\right)}^{\frac{1}{2}}$

$6\times {\left(6\right)}^{\frac{1}{2}}\times {\left(2\right)}^{\frac{1}{2}}=6\times {\left(6\times 2\right)}^{\frac{1}{2}}=6\times {\left(4\times 3\right)}^{\frac{1}{2}}=6\times {\left(4\right)}^{\frac{1}{2}}\times {\left(3\right)}^{\frac{1}{2}}$

$6\times {\left(4\right)}^{\frac{1}{2}}\times {\left(3\right)}^{\frac{1}{2}}=6\times {\left(2^2\right)}^{\frac{1}{2}}\times {\left(3\right)}^{\frac{1}{2}}=6\times \left|2\right|\times \sqrt{3}=12\sqrt{3}$