Question

question
express as a fraction in simplest form with a rational denominator:
$\frac{-6}{-3+sqrt{6}}$

Explanation:

Step1: Identify conjugate of denominator

The conjugate of $-3+\sqrt{6}$ is $-3-\sqrt{6}$.

Step2: Multiply numerator and denominator by conjugate

$$\frac{-6}{-3+\sqrt{6}} \times \frac{-3-\sqrt{6}}{-3-\sqrt{6}} = \frac{-6(-3-\sqrt{6})}{(-3)^2 - (\sqrt{6})^2}$$

Step3: Simplify numerator and denominator

Numerator: $-6(-3-\sqrt{6}) = 18 + 6\sqrt{6}$
Denominator: $9 - 6 = 3$
$$\frac{18 + 6\sqrt{6}}{3}$$

Step4: Divide terms by common factor 3

$$\frac{18}{3} + \frac{6\sqrt{6}}{3} = 6 + 2\sqrt{6}$$

Answer:

$6 + 2\sqrt{6}$