Question

rationalize the denominator.

\\\frac{\sqrt{3} - 8\sqrt{2}}{3\sqrt{3} + 5\sqrt{2}}\\

\\\frac{\sqrt{3} - 8\sqrt{2}}{3\sqrt{3} + 5\sqrt{2}} = \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

$$ \frac{\sqrt{3} - 8\sqrt{2}}{3\sqrt{3} + 5\sqrt{2}} \cdot \frac{3\sqrt{3} - 5\sqrt{2}}{3\sqrt{3} - 5\sqrt{2}} $$

Step 2: Simplify the denominator

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

$$ = 9(3) - 25(2) = 27 - 50 = -23 $$

Step 3: Expand the numerator

$$ (\sqrt{3} - 8\sqrt{2})(3\sqrt{3} - 5\sqrt{2}) = \sqrt{3}(3\sqrt{3}) - \sqrt{3}(5\sqrt{2}) - 8\sqrt{2}(3\sqrt{3}) + 8\sqrt{2}(5\sqrt{2}) $$

$$ = 3(3) - 5\sqrt{6} - 24\sqrt{6} + 40(2) $$

$$ = 9 - 29\sqrt{6} + 80 = 89 - 29\sqrt{6} $$

Step 4: Combine and simplify

$$ \frac{89 - 29\sqrt{6}}{-23} = \frac{-89 + 29\sqrt{6}}{23} $$

Answer:

$$ \frac{-89 + 29\sqrt{6}}{23} $$