Chapter 11.9, Problem 21WE

To determine

Simplify.

Answer to Problem 21WE

Simplified form of $\left(4\sqrt{11}-2\sqrt{2}\right)\left(6\sqrt{11}+8\sqrt{2}\right)$ is $232+20\sqrt{22}$ .

Explanation of Solution

Given information:

The expression $\left(4\sqrt{11}-2\sqrt{2}\right)\left(6\sqrt{11}+8\sqrt{2}\right)$ .

  $\begin{array}{l}\left(4\sqrt{11}-2\sqrt{2}\right)\left(6\sqrt{11}+8\sqrt{2}\right)\\ =4\sqrt{11}\left(6\sqrt{11}+8\sqrt{2}\right)-2\sqrt{2}\left(6\sqrt{11}+8\sqrt{2}\right)\\ =4\sqrt{11}\cdot 6\sqrt{11}+4\sqrt{11}\cdot 8\sqrt{2}-2\sqrt{2}\cdot 6\sqrt{11}-2\sqrt{2}\cdot 8\sqrt{2}\\ =24\cdot 11+32\sqrt{11\cdot 2}-12\sqrt{11\cdot 2}-16\cdot 2\end{array}$

  $\begin{array}{l}=24\cdot 11+32\sqrt{11\cdot 2}-12\sqrt{11\cdot 2}-16\cdot 2\\ =264+32\sqrt{22}-12\sqrt{22}-32\\ =264+20\sqrt{22}-32\\ =\boxed{232+20\sqrt{22}}\end{array}$

Simplified form of $\left(4\sqrt{11}-2\sqrt{2}\right)\left(6\sqrt{11}+8\sqrt{2}\right)$ is $232+20\sqrt{22}$ .