Question

In the following exercises, simplify.

$\frac{6}{3-\sqrt{7}}$

Short answer

The simplified form is$3\left(3+\sqrt{7}\right)$.

Step-by-step solution

Step 1. Given information.

The given expression is,

$\frac{6}{3-\sqrt{7}}$.

Step 2. Simplifying.

Now, to rationalize we will multiply the numerator and denominator by the conjugate of the denominator and simplify.

$$\begin{gathered} \frac{6}{3-\sqrt{7}}=\frac{6\left(3+\sqrt{7}\right)}{\left(3-\sqrt{7}\right)\left(3+\sqrt{7}\right)} \\ =\frac{6\left(3+\sqrt{7}\right)}{3^2-{\left(\sqrt{7}\right)}^2} \\ =\frac{6\left(3+\sqrt{7}\right)}{9-7} \\ =\frac{6\left(3+\sqrt{7}\right)}{2} \\ =3\left(3+\sqrt{7}\right) \end{gathered}$$