Question

rationalize the denominator. simplify if possible.

\\\frac{\sqrt{5} + 8}{\sqrt{5} - \sqrt{3}}\\

\\\frac{\sqrt{5} + 8}{\sqrt{5} - \sqrt{3}} = \square\\
(simplify your answer. type an exact answer using radicals as needed.)

Explanation:

⚡ Using what you learned: Rationalizing Denominators

Step 1: Multiply by the conjugate

$$ \frac{\sqrt{5} + 8}{\sqrt{5} - \sqrt{3}} \cdot \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}} $$

Step 2: Expand the numerator and denominator

Numerator:

$$ (\sqrt{5} + 8)(\sqrt{5} + \sqrt{3}) = 5 + \sqrt{15} + 8\sqrt{5} + 8\sqrt{3} $$

Denominator:

$$ (\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3}) = 5 - 3 = 2 $$

Step 3: Combine into a single fraction

$$ \frac{5 + 8\sqrt{5} + 8\sqrt{3} + \sqrt{15}}{2} $$

Answer:

$$ \frac{5 + 8\sqrt{5} + 8\sqrt{3} + \sqrt{15}}{2} $$