Question

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

Explanation:

Step1: Multiply by conjugate

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

Step2: Expand denominator

Use $(a + b)(a - b)=a^{2}-b^{2}$. Here $a=-10$, $b = \sqrt{5}$.
$\frac{10(-10 - \sqrt{5})}{(-10)^{2}-(\sqrt{5})^{2}}=\frac{10(-10 - \sqrt{5})}{100 - 5}$

Step3: Simplify denominator and distribute numerator

$\frac{-100-10\sqrt{5}}{95}=\frac{-20 - 2\sqrt{5}}{19}$

Answer:

$\frac{-20 - 2\sqrt{5}}{19}$