To simplify the expression \(\frac{5}{-\sqrt{2} + \sqrt{7}}\) and rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(-\sqrt{2} - \sqrt{7}\).
So, we perform the following steps:
\[ \frac{5}{-\sqrt{2} + \sqrt{7}} \cdot \frac{-\sqrt{2} - \sqrt{7}}{-\sqrt{2} - \sqrt{7}} = \frac{5(-\sqrt{2} - \sqrt{7})}{(-\sqrt{2} + \sqrt{7})(-\sqrt{2} - \sqrt{7})} \]
Now, we simplify the denominator:
\[ (-\sqrt{2} + \sqrt{7})(-\sqrt{2} - \sqrt{7}) = (-\sqrt{2})^2 - (\sqrt{7})^2 = 2 - 7 = -5 \]
Now we can substitute back into the expression:
\[ \frac{5(-\sqrt{2} - \sqrt{7})}{-5} \]
The \(-5\) in the numerator and denominator cancels out:
\[ =\sqrt{2} + \sqrt{7} \]
Thus, the simplified expression is:
\[ \sqrt{2} + \sqrt{7} \]
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