Question

question
express as a fraction in simplest form with a rational denominator:
$\frac{9}{10+sqrt{5}}$

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $10-\sqrt{5}$:
$$\frac{9}{10+\sqrt{5}} \times \frac{10-\sqrt{5}}{10-\sqrt{5}} = \frac{9(10-\sqrt{5})}{(10+\sqrt{5})(10-\sqrt{5})}$$

Step2: Expand denominator via difference of squares

Use $(a+b)(a-b)=a^2-b^2$ for denominator:
$$\frac{9(10-\sqrt{5})}{10^2 - (\sqrt{5})^2} = \frac{90 - 9\sqrt{5}}{100 - 5}$$

Step3: Simplify denominator

Calculate denominator value:
$$\frac{90 - 9\sqrt{5}}{95}$$

Step4: Check for common factors

No shared factors in numerator/denominator, so this is simplest form.

Answer:

$\frac{90 - 9\sqrt{5}}{95}$