Chapter 7.4, Problem 5AGP

To determine

To simplify the given expression $\frac{3}{2+\sqrt{2}}$ .

Answer to Problem 5AGP

  $3-\frac{3}{\sqrt{2}}$

Explanation of Solution

Given : $\frac{3}{2+\sqrt{2}}$

Concept used:

• Distributive law: It states that the product of sum of two number and a third number is equal to the sum of the products of third number and the two numbers separately. For examples a(b + c) = ab + ac and vice versa.

  $·\left(a-b\right)\left(a+b\right)=a^2-b^2$

Calculation:

  $\begin{array}{l}=\frac{3}{2+\sqrt{2}}\\ =\frac{3}{2+\sqrt{2}}\times \frac{2-\sqrt{2}}{2-\sqrt{2}}\end{array}$ (multiplying numerator and denominator with same number as shown)

  $\begin{array}{l}=\frac{3\left(2-\sqrt{2}\right)}{\left(2+\sqrt{2}\right)\left(2-\sqrt{2}\right)}\\ =\frac{3·2-3·\sqrt{2}}{{\left(2\right)}^2-{\left(\sqrt{2}\right)}^2}\\ =\frac{6-3\sqrt{2}}{4-2}\\ =\frac{6-3\sqrt{2}}{2}\\ =\frac{6}{2}-\frac{3\sqrt{2}}{2}\\ =3-\frac{3}{\sqrt{2}}\end{array}$

Thus, obtained form is the simplified form of the given expression.