Chapter 7.4, Problem 14CYU

To determine

To simplify the given expression $\frac{1}{4+\sqrt{12}}$ .

Answer to Problem 14CYU

  $1-\frac{1}{2}\sqrt{3}$

Explanation of Solution

Given : $\frac{1}{4+\sqrt{12}}$

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:

  $=\frac{1}{4+\sqrt{12}}$

(multiplying numerator and denominator with $\left(4-\sqrt{12}\right)$ )

  $\begin{array}{l}=\frac{1}{4+\sqrt{12}}\times \frac{4-\sqrt{12}}{4-\sqrt{12}}\\ =\frac{1\left(4-\sqrt{12}\right)}{\left(4+\sqrt{12}\right)\left(4-\sqrt{12}\right)}\\ =\frac{4-\sqrt{12}}{{\left(4\right)}^2-{\left(\sqrt{12}\right)}^2}\\ =\frac{4-\sqrt{2\times 2\times 3}}{16-12}\\ =\frac{4-2\sqrt{3}}{4}\\ =\frac{4}{4}-\frac{2}{4}\sqrt{3}\\ =1-\frac{1}{2}\sqrt{3}\end{array}$

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