Question

simplify. rationalize the denominator.\\(\frac{9}{-5 + \sqrt{3}}\\)

Explanation:

Step1: Identify conjugate

The denominator is $-5 + \sqrt{3}$, its conjugate is $-5 - \sqrt{3}$. Multiply numerator and denominator by this conjugate.
$$\frac{9}{-5 + \sqrt{3}} \times \frac{-5 - \sqrt{3}}{-5 - \sqrt{3}}$$

Step2: Multiply numerators and denominators

Numerator: $9 \times (-5 - \sqrt{3}) = -45 - 9\sqrt{3}$
Denominator: Use $(a + b)(a - b) = a^2 - b^2$, where $a = -5$, $b = \sqrt{3}$.
$$(-5)^2 - (\sqrt{3})^2 = 25 - 3 = 22$$

Step3: Simplify the fraction

The expression becomes $\frac{-45 - 9\sqrt{3}}{22}$, which can be factored as $\frac{-9(5 + \sqrt{3})}{22}$ or written as $-\frac{45 + 9\sqrt{3}}{22}$.

Answer:

$\boldsymbol{-\frac{45 + 9\sqrt{3}}{22}}$ (or equivalent forms like $\frac{-45 - 9\sqrt{3}}{22}$)