Chapter 7.4, Problem 12CYU

To determine

To simplify the given expression $\frac{5}{2-\sqrt{6}}$ .

Answer to Problem 12CYU

  $-5-\frac{5}{2}\sqrt{6}$

Explanation of Solution

Given : $\frac{5}{2-\sqrt{6}}$

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:

  $=\frac{5}{2-\sqrt{6}}$

(multiplying numerator and denominator with $\left(2+\sqrt{6}\right)$ )

  $\begin{array}{l}=\frac{5}{2-\sqrt{6}}\times \frac{2+\sqrt{6}}{2+\sqrt{6}}\\ =\frac{5\left(2+\sqrt{6}\right)}{\left(2-\sqrt{6}\right)\left(2+\sqrt{6}\right)}\\ =\frac{5·2+5·\sqrt{6}}{{\left(2\right)}^2-{\left(\sqrt{6}\right)}^2}=\frac{10+5\sqrt{6}}{4-6}\\ =\frac{10+5\sqrt{6}}{-2}\\ =-\frac{10}{2}-\frac{5}{2}\sqrt{6}\\ =-5-\frac{5}{2}\sqrt{6}\\ \end{array}$

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