Chapter 6.4, Problem 6WE

To determine

To Simplify:

  $\left(6-\sqrt{3}\right)\left(4+\sqrt{3}\right)$

Answer to Problem 6WE

  $21+2\sqrt{3}$

Explanation of Solution

Given:

The expression is :

  $\left(6-\sqrt{3}\right)\left(4+\sqrt{3}\right)$

Calculation:

  $\left(6-\sqrt{3}\right)\left(4+\sqrt{3}\right)$

Use FOIL method:

F represents the multiplication of first terms of each binomial = $\text{6}\cdot 4=24$

O represents the multiplication of the outermost terms of the product = $\text{6}\cdot \sqrt{3}=6\sqrt{3}$

I represents the multiplication of the innermost terms of the product = $\left(-\sqrt{3}\right)\cdot 4=-4\sqrt{3}$

L represents the multiplication of the last terms of each binomial = $\left(-\sqrt{3}\right)\cdot \sqrt{3}=-3$

Add all the products:

  $\begin{array}{l}24+6\sqrt{3}-4\sqrt{3}-3\\ =21+2\sqrt{3}\mathrm{............}[\text{add like terms]}\\ \text{So, }\\ \left(6-\sqrt{3}\right)\left(4+\sqrt{3}\right)=21+2\sqrt{3}\end{array}$