Question

question 13 (3 points)
rationalize

\\\frac{18 - \sqrt{6}}{3 - \sqrt{6}}\\

Explanation:

Multiply by the conjugate of the denominator

$$ \frac{18 - \sqrt{6}}{3 - \sqrt{6}} \cdot \frac{3 + \sqrt{6}}{3 + \sqrt{6}} $$

Expand the numerator and denominator

$$ \text{Numerator: } (18 - \sqrt{6})(3 + \sqrt{6}) = 54 + 18\sqrt{6} - 3\sqrt{6} - 6 = 48 + 15\sqrt{6} $$

$$ \text{Denominator: } (3 - \sqrt{6})(3 + \sqrt{6}) = 9 - 6 = 3 $$

Simplify the fraction

$$ \frac{48 + 15\sqrt{6}}{3} = \frac{48}{3} + \frac{15\sqrt{6}}{3} = 16 + 5\sqrt{6} $$

Answer:

\(16 + 5\sqrt{6}\)