Chapter 6.4, Problem 2WE

To determine

To Simplify:

  $\left(5+\sqrt{2}\right)\left(5-\sqrt{2}\right)$

Answer to Problem 2WE

23

Explanation of Solution

Given:

The expression is :

  $\left(5+\sqrt{2}\right)\left(5-\sqrt{2}\right)$

Calculation:

  $\left(5+\sqrt{2}\right)\left(5-\sqrt{2}\right)$

Use FOIL method:

F represents the multiplication of first terms of each binomial = $\text{5}\cdot 5=5^2=25$

O represents the multiplication of the outermost terms of the product = $\text{5}\cdot \left(-\sqrt{2}\right)=-5\sqrt{2}$

I represents the multiplication of the innermost terms of the product = $\sqrt{2}\cdot 5=5\sqrt{2}$

L represents the multiplication of the last terms of each binomial = $\sqrt{2}\cdot \left(-\sqrt{2}\right)=-2$

Add all the products:

  $\begin{array}{l}25-5\sqrt{2}+5\sqrt{2}-2\\ =25-2\\ =2\\ \text{So, }\\ \left(5+\sqrt{2}\right)\left(5-\sqrt{2}\right)=23\end{array}$