Question

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simplify. rationalize the denominator.
\\(\frac{-10}{8 + \sqrt{5}}\\)

Explanation:

Step1: Multiply by conjugate

Multiply numerator and denominator by $8 - \sqrt{5}$:
$\frac{-10}{8+\sqrt{5}} \times \frac{8-\sqrt{5}}{8-\sqrt{5}} = \frac{-10(8-\sqrt{5})}{(8+\sqrt{5})(8-\sqrt{5})}$

Step2: Expand denominator

Use difference of squares $(a+b)(a-b)=a^2-b^2$:
$\frac{-10(8-\sqrt{5})}{8^2 - (\sqrt{5})^2} = \frac{-10(8-\sqrt{5})}{64 - 5}$

Step3: Simplify denominator

Calculate denominator value:
$\frac{-10(8-\sqrt{5})}{59}$

Step4: Distribute numerator

Expand the numerator:
$\frac{-80 + 10\sqrt{5}}{59}$ or $\frac{10\sqrt{5} - 80}{59}$

Answer:

$\frac{10\sqrt{5} - 80}{59}$