To simplify and rationalize the expression \(-\frac{8}{5 - \sqrt{5}}\), we will multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(5 - \sqrt{5}\) is \(5 + \sqrt{5}\).
Let's perform the multiplication:
\[ -\frac{8}{5 - \sqrt{5}} \cdot \frac{5 + \sqrt{5}}{5 + \sqrt{5}} = -\frac{8(5 + \sqrt{5})}{(5 - \sqrt{5})(5 + \sqrt{5})} \]
Now, we need to calculate the denominator, which is a difference of squares:
\[ (5 - \sqrt{5})(5 + \sqrt{5}) = 5^2 - (\sqrt{5})^2 = 25 - 5 = 20 \]
Now substitute this back into the expression:
\[ -\frac{8(5 + \sqrt{5})}{20} \]
We can simplify the fraction:
\[ -\frac{8(5 + \sqrt{5})}{20} = -\frac{2(5 + \sqrt{5})}{5} \]
Now distribute the -2 in the numerator:
\[ -\frac{10 + 2\sqrt{5}}{5} \]
So, the final simplified expression is:
\[ -\frac{10 + 2\sqrt{5}}{5} \]
This is the rationalized and simplified form of the original expression.
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