Chapter 7.4, Problem 11CYU

To determine

To simplify the given expression $\frac{3}{3+\sqrt{5}}$ .

Answer to Problem 11CYU

  $\frac{9}{4}-\frac{3}{4}\sqrt{5}$

Explanation of Solution

Given : $\frac{3}{3+\sqrt{5}}$

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:

  $=\frac{3}{3+\sqrt{5}}$

(multiplying numerator and denominator with $\left(3-\sqrt{5}\right)$ )

  $\begin{array}{l}=\frac{3}{3+\sqrt{5}}\times \frac{3-\sqrt{5}}{3-\sqrt{5}}\\ =\frac{3\left(3-\sqrt{5}\right)}{\left(3+\sqrt{5}\right)\left(3-\sqrt{5}\right)}\\ =\frac{3·3-3·\sqrt{5}}{{\left(3\right)}^2-{\left(\sqrt{5}\right)}^2}\\ =\frac{9-3\sqrt{5}}{9-5}\\ =\frac{9-3\sqrt{5}}{4}\\ =\frac{9}{4}-\frac{3}{4}\sqrt{5}\end{array}$

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