Question

rationalize the denominator.

\\\frac{\sqrt{5}}{\sqrt{10} - \sqrt{5}}\\

\\\frac{\sqrt{5}}{\sqrt{10} - \sqrt{5}} = \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

$$ \frac{\sqrt{5}}{\sqrt{10} - \sqrt{5}} \cdot \frac{\sqrt{10} + \sqrt{5}}{\sqrt{10} + \sqrt{5}} $$

Step 2: Simplify the denominator

$$ (\sqrt{10} - \sqrt{5})(\sqrt{10} + \sqrt{5}) = 10 - 5 = 5 $$

Step 3: Simplify the numerator

$$ \sqrt{5}(\sqrt{10} + \sqrt{5}) = \sqrt{50} + 5 = 5\sqrt{2} + 5 $$

Step 4: Divide terms

$$ \frac{5\sqrt{2} + 5}{5} = \sqrt{2} + 1 $$

Answer:

$$ \sqrt{2} + 1 $$