Chapter 7.4, Problem 16CYU

To determine

To simplify the given expression $\frac{6}{5+\sqrt{11}}$ .

Answer to Problem 16CYU

  $\frac{15}{7}-\frac{3}{7}\sqrt{11}$

Explanation of Solution

Given : $\frac{6}{5+\sqrt{11}}$

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:

  $=\frac{6}{5+\sqrt{11}}$

(multiplying numerator and denominator with $\left(5-\sqrt{11}\right)$

  $\begin{array}{l}=\frac{6}{5+\sqrt{11}}\times \frac{5-\sqrt{11}}{5-\sqrt{11}}\\ =\frac{6\left(5-\sqrt{11}\right)}{\left(5+\sqrt{11}\right)\left(5-\sqrt{11}\right)}\\ =\frac{6·5-6·\sqrt{11}}{{\left(5\right)}^2-{\left(\sqrt{11}\right)}^2}\\ =\frac{30-6\sqrt{11}}{25-11}\\ =\frac{30-6\sqrt{11}}{14}\\ =\frac{30}{14}-\frac{6}{14}\sqrt{11}\\ =\frac{15}{7}-\frac{3}{7}\sqrt{11}\end{array}$

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