Evaluate
\sqrt{10}-\sqrt{5}\approx 0.926209683
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\frac{5\left(\sqrt{10}-\sqrt{5}\right)}{\left(\sqrt{10}+\sqrt{5}\right)\left(\sqrt{10}-\sqrt{5}\right)}
Rationalize the denominator of \frac{5}{\sqrt{10}+\sqrt{5}} by multiplying numerator and denominator by \sqrt{10}-\sqrt{5}.
\frac{5\left(\sqrt{10}-\sqrt{5}\right)}{\left(\sqrt{10}\right)^{2}-\left(\sqrt{5}\right)^{2}}
Consider \left(\sqrt{10}+\sqrt{5}\right)\left(\sqrt{10}-\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
\frac{5\left(\sqrt{10}-\sqrt{5}\right)}{10-5}
Square \sqrt{10}. Square \sqrt{5}.
\frac{5\left(\sqrt{10}-\sqrt{5}\right)}{5}
Subtract 5 from 10 to get 5.
\sqrt{10}-\sqrt{5}
Cancel out 5 and 5.
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