Chapter 11.9, Problem 20WE

To determine

Simplify.

Answer to Problem 20WE

Simplified form of $\left(7\sqrt{13}+2\sqrt{6}\right)\left(2\sqrt{13}+3\sqrt{6}\right)$ is $-12-16\sqrt{35}$ .

Explanation of Solution

Given information:

The expression $\left(7\sqrt{13}+2\sqrt{6}\right)\left(2\sqrt{13}+3\sqrt{6}\right)$ .

  $\begin{array}{l}\left(7\sqrt{13}+2\sqrt{6}\right)\left(2\sqrt{13}+3\sqrt{6}\right)\\ =7\sqrt{13}\left(2\sqrt{13}+3\sqrt{6}\right)+2\sqrt{6}\left(2\sqrt{13}+3\sqrt{6}\right)\\ =7\sqrt{13}\cdot 2\sqrt{13}+7\sqrt{13}\cdot 3\sqrt{6}+2\sqrt{6}\cdot 2\sqrt{13}+2\sqrt{6}\cdot 3\sqrt{6}\\ =14\cdot 13+21\cdot \sqrt{13\cdot 6}+4\cdot \sqrt{13\cdot 6}+6\cdot 6\end{array}$

  $\begin{array}{l}=14\cdot 13+21\cdot \sqrt{13\cdot 6}+4\cdot \sqrt{13\cdot 6}+6\cdot 6\\ =182+21\sqrt{78}+4\sqrt{78}+36\\ =218+25\sqrt{78}\end{array}$

Simplified form of $\left(7\sqrt{13}+2\sqrt{6}\right)\left(2\sqrt{13}+3\sqrt{6}\right)$ is $218+25\sqrt{78}$ .