Question

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simplify. rationalize the denominator.
\\(\frac{-8}{6 - \sqrt{2}}\\)

Explanation:

Step1: Multiply by conjugate

Multiply numerator and denominator by $6+\sqrt{2}$:
$\frac{-8}{6-\sqrt{2}} \times \frac{6+\sqrt{2}}{6+\sqrt{2}} = \frac{-8(6+\sqrt{2})}{(6-\sqrt{2})(6+\sqrt{2})}$

Step2: Expand denominator

Use difference of squares $(a-b)(a+b)=a^2-b^2$:
$\frac{-8(6+\sqrt{2})}{6^2 - (\sqrt{2})^2} = \frac{-8(6+\sqrt{2})}{36-2}$

Step3: Simplify denominator

Calculate denominator value:
$\frac{-8(6+\sqrt{2})}{34}$

Step4: Simplify the fraction

Divide numerator and denominator by 2:
$\frac{-4(6+\sqrt{2})}{17} = \frac{-24 - 4\sqrt{2}}{17}$

Answer:

$\frac{-24 - 4\sqrt{2}}{17}$ (or $\frac{-(24 + 4\sqrt{2})}{17}$)