Question

simplify. rationalize the denominator.\\(\frac{9}{-7 - \sqrt{3}}\\)

Explanation:

Step1: Identify conjugate of denominator

The conjugate of $-7 - \sqrt{3}$ is $-7 + \sqrt{3}$.

Step2: Multiply numerator and denominator by conjugate

$$\frac{9}{-7 - \sqrt{3}} \times \frac{-7 + \sqrt{3}}{-7 + \sqrt{3}} = \frac{9(-7 + \sqrt{3})}{(-7)^2 - (\sqrt{3})^2}$$

Step3: Simplify denominator

$$(-7)^2 - (\sqrt{3})^2 = 49 - 3 = 46$$

Step4: Simplify numerator

$$9(-7 + \sqrt{3}) = -63 + 9\sqrt{3}$$

Step5: Write final simplified form

$$\frac{-63 + 9\sqrt{3}}{46} = -\frac{63 - 9\sqrt{3}}{46}$$

Answer:

$\boldsymbol{-\frac{63 - 9\sqrt{3}}{46}}$ (or $\boldsymbol{\frac{-63 + 9\sqrt{3}}{46}}$)