Question

question
express as a fraction in simplest form with a rational denominator:
$\frac{-9}{6 - sqrt{5}}$

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $6+\sqrt{5}$ to rationalize:
$\frac{-9}{6-\sqrt{5}} \times \frac{6+\sqrt{5}}{6+\sqrt{5}} = \frac{-9(6+\sqrt{5})}{(6-\sqrt{5})(6+\sqrt{5})}$

Step2: Expand denominator (difference of squares)

Use $(a-b)(a+b)=a^2-b^2$:
$\frac{-9(6+\sqrt{5})}{6^2 - (\sqrt{5})^2} = \frac{-9(6+\sqrt{5})}{36 - 5}$

Step3: Simplify denominator

Calculate denominator value:
$\frac{-9(6+\sqrt{5})}{31}$

Step4: Distribute numerator

Expand the numerator:
$\frac{-54 - 9\sqrt{5}}{31}$

Answer:

$\frac{-54 - 9\sqrt{5}}{31}$ (or equivalently $\frac{-(54 + 9\sqrt{5})}{31}$)