Question

simplify. rationalize the denominator.
\frac{\sqrt{5}}{-10 + \sqrt{7}}

Explanation:

Step1: Multiply by conjugate

Multiply numerator and denominator by $- 10-\sqrt{7}$.
$\frac{\sqrt{5}}{-10 + \sqrt{7}}\times\frac{-10-\sqrt{7}}{-10-\sqrt{7}}$

Step2: Expand numerator and denominator

Numerator: $\sqrt{5}\times(-10-\sqrt{7})=-10\sqrt{5}-\sqrt{35}$.
Denominator: $(-10 + \sqrt{7})(-10-\sqrt{7})=(-10)^2-(\sqrt{7})^2 = 100 - 7=93$.

Answer:

$\frac{-10\sqrt{5}-\sqrt{35}}{93}$