Chapter 6.4, Problem 5WE

To determine

To Simplify:

  $\left(1+\sqrt{2}\right)\left(3+\sqrt{2}\right)$

Answer to Problem 5WE

  $5+4\sqrt{2}$

Explanation of Solution

Given:

The expression is :

  $\left(1+\sqrt{2}\right)\left(3+\sqrt{2}\right)$

Calculation:

  $\left(1+\sqrt{2}\right)\left(3+\sqrt{2}\right)$

Use FOIL method:

F represents the multiplication of first terms of each binomial = $\text{1}\cdot 3=3$

O represents the multiplication of the outermost terms of the product = $\text{1}\cdot \sqrt{2}=\sqrt{2}$

I represents the multiplication of the innermost terms of the product = $\sqrt{2}\cdot 3=3\sqrt{2}$

L represents the multiplication of the last terms of each binomial = $\sqrt{\text{2}}\cdot \sqrt{2}=2$

Add all the products:

  $\begin{array}{l}3+\sqrt{2}+3\sqrt{2}+2\\ =5+4\sqrt{2}\mathrm{............}[\text{add like terms]}\\ \text{So, }\\ \left(1+\sqrt{2}\right)\left(3+\sqrt{2}\right)=5+4\sqrt{2}\end{array}$