Chapter 7.4, Problem 15CYU

To determine

To simplify the given expression $\frac{4}{6-\sqrt{7}}$ .

Answer to Problem 15CYU

  $\frac{24}{29}+\frac{4}{29}\sqrt{7}$

Explanation of Solution

Given : $\frac{4}{6-\sqrt{7}}$

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:

  $=\frac{4}{6-\sqrt{7}}$

(multiplying numerator and denominator with $\left(6+\sqrt{7}\right)$

  $\begin{array}{l}=\frac{4}{6-\sqrt{7}}\times \frac{6+\sqrt{7}}{6+\sqrt{7}}\\ =\frac{4\left(6+\sqrt{7}\right)}{\left(6-\sqrt{7}\right)\left(6+\sqrt{7}\right)}\\ =\frac{4·6+4·\sqrt{7}}{{\left(6\right)}^2-{\left(\sqrt{7}\right)}^2}\\ =\frac{24+4\sqrt{7}}{36-7}\\ =\frac{24+4\sqrt{7}}{29}\\ =\frac{24}{29}+\frac{4}{29}\sqrt{7}\end{array}$

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