Evaluate
5\left(\sqrt{10}-\sqrt{5}\right)\approx 4.631048413
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\frac{25\left(\sqrt{10}-\sqrt{5}\right)}{\left(\sqrt{10}+\sqrt{5}\right)\left(\sqrt{10}-\sqrt{5}\right)}
Rationalize the denominator of \frac{25}{\sqrt{10}+\sqrt{5}} by multiplying numerator and denominator by \sqrt{10}-\sqrt{5}.
\frac{25\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{25\left(\sqrt{10}-\sqrt{5}\right)}{10-5}
Square \sqrt{10}. Square \sqrt{5}.
\frac{25\left(\sqrt{10}-\sqrt{5}\right)}{5}
Subtract 5 from 10 to get 5.
5\left(\sqrt{10}-\sqrt{5}\right)
Divide 25\left(\sqrt{10}-\sqrt{5}\right) by 5 to get 5\left(\sqrt{10}-\sqrt{5}\right).
5\sqrt{10}-5\sqrt{5}
Use the distributive property to multiply 5 by \sqrt{10}-\sqrt{5}.
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