Chapter 11.9, Problem 15WE

To determine

Simplify.

Answer to Problem 15WE

The simplified form of $\left(6\sqrt{5}-\sqrt{7}\right)\left(6\sqrt{5}+\sqrt{7}\right)$ is $12$ .

Explanation of Solution

Given information:

The expression $\left(6\sqrt{5}-\sqrt{7}\right)\left(6\sqrt{5}+\sqrt{7}\right)$ .

Formula used:

  $\left(a+b\right)\left(a-b\right)=a^2-b^2$

  $\left(6\sqrt{5}-\sqrt{7}\right)\left(6\sqrt{5}+\sqrt{7}\right)$

As the expression resembles $\left(a+b\right)\left(a-b\right)$ ,

we will use identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$

  $\begin{array}{l}\left(6\sqrt{5}-\sqrt{7}\right)\left(6\sqrt{5}+\sqrt{7}\right)\\ ={\left(6\sqrt{5}\right)}^2-{\left(\sqrt{7}\right)}^2\\ =36\cdot 5-7\\ =180-7\end{array}$

  $=\boxed{173}$

The simplified form of $\left(3\sqrt{5}-\sqrt{2}\right)\left(3\sqrt{5}+\sqrt{2}\right)$ is $173$ .