Question

rationalize the denominator.

\\\frac{84}{6\sqrt{7} + 7\sqrt{6}}\\

\\\frac{84}{6\sqrt{7} + 7\sqrt{6}} = \square\\
(simplify your answer. type an exact answer, us

Explanation:

⚡ Using what you learned: Rationalizing Denominators

Step 1: Multiply by the conjugate

$$ \frac{84}{6\sqrt{7} + 7\sqrt{6}} \cdot \frac{6\sqrt{7} - 7\sqrt{6}}{6\sqrt{7} - 7\sqrt{6}} = \frac{84(6\sqrt{7} - 7\sqrt{6})}{(6\sqrt{7})^2 - (7\sqrt{6})^2} $$

Step 2: Simplify the denominator

$$ (6\sqrt{7})^2 = 36 \cdot 7 = 252 $$

$$ (7\sqrt{6})^2 = 49 \cdot 6 = 294 $$

$$ 252 - 294 = -42 $$

Step 3: Divide and simplify

$$ \frac{84(6\sqrt{7} - 7\sqrt{6})}{-42} = -2(6\sqrt{7} - 7\sqrt{6}) $$

$$ -12\sqrt{7} + 14\sqrt{6} = 14\sqrt{6} - 12\sqrt{7} $$

Answer:

\(14\sqrt{6} - 12\sqrt{7}\)