Chapter 11.9, Problem 13WE

To determine

Simplify.

Answer to Problem 13WE

The simplified form of $\left(2\sqrt{7}+\sqrt{3}\right)\left(2\sqrt{7}-\sqrt{3}\right)$ is $12$ .

Explanation of Solution

Given information:

The expression $\left(2\sqrt{7}+\sqrt{3}\right)\left(2\sqrt{7}-\sqrt{3}\right)$ .

Formula used:

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

  $\left(2\sqrt{7}+\sqrt{3}\right)\left(2\sqrt{7}-\sqrt{3}\right)$

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

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

  $\begin{array}{l}\left(2\sqrt{7}+\sqrt{3}\right)\left(2\sqrt{7}-\sqrt{3}\right)\\ ={\left(2\sqrt{7}\right)}^2-{\left(\sqrt{3}\right)}^2\\ =4\cdot 7-3\\ =28-3\end{array}$

  $=\boxed{25}$

The simplified form of $\left(2\sqrt{7}+\sqrt{3}\right)\left(2\sqrt{7}-\sqrt{3}\right)$ is $25$ .