Chapter 7.4, Problem 13CYU

To determine

To simplify the given expression $\frac{2}{1-\sqrt{10}}$ .

Answer to Problem 13CYU

  $-\frac{2}{9}-\frac{2}{9}\sqrt{10}$

Explanation of Solution

Given : $\frac{2}{1-\sqrt{10}}$

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:

  $=\frac{2}{1-\sqrt{10}}$

(multiplying numerator and denominator with $\left(1+\sqrt{10}\right)$ )

  $\begin{array}{l}=\frac{2}{1-\sqrt{10}}\times \frac{1+\sqrt{10}}{1+\sqrt{10}}\\ =\frac{2\left(1+\sqrt{10}\right)}{\left(1-\sqrt{10}\right)\left(1+\sqrt{10}\right)}\\ =\frac{2·1+2·\sqrt{10}}{{\left(1\right)}^2-{\left(\sqrt{10}\right)}^2}\\ =\frac{2+2\sqrt{10}}{1-10}\\ =\frac{2+2\sqrt{10}}{-9}\\ =\frac{-2}{9}-\frac{2}{9}\sqrt{10}\end{array}$

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