Question

rationalize the denominator.

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

\\\frac{\sqrt{3} - 2\sqrt{2}}{5\sqrt{3} + 8\sqrt{2}} = \square\\
(simplify your answer. type an exact answer, using

Explanation:

⚡ Using what you learned: Rationalizing Denominators

Step 1: Multiply by the conjugate

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

Step 2: Expand the numerator

$$ (\sqrt{3} - 2\sqrt{2})(5\sqrt{3} - 8\sqrt{2}) = 5(3) - 8\sqrt{6} - 10\sqrt{6} + 16(2) $$

$$ = 15 - 18\sqrt{6} + 32 $$

$$ = 47 - 18\sqrt{6} $$

Step 3: Expand the denominator

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

$$ = 25(3) - 64(2) $$

$$ = 75 - 128 $$

$$ = -53 $$

Step 4: Simplify the fraction

$$ \frac{47 - 18\sqrt{6}}{-53} = \frac{-47 + 18\sqrt{6}}{53} $$

Answer:

$$ \frac{18\sqrt{6} - 47}{53} $$