Solve 8/sqrt{10-sqrt{2}}

$$\frac { 8 } { \sqrt { 10 } - \sqrt { 2 } }$$

Evaluate

$$\sqrt{2}+\sqrt{10}\approx 4.576491223$$

Solution Steps

Rationalize the denominator of \frac{8}{\sqrt{10}-\sqrt{2}} by multiplying numerator and denominator by \sqrt{10}+\sqrt{2}.

$$\frac{8\left(\sqrt{10}+\sqrt{2}\right)}{\left(\sqrt{10}-\sqrt{2}\right)\left(\sqrt{10}+\sqrt{2}\right)}$$

Consider \left(\sqrt{10}-\sqrt{2}\right)\left(\sqrt{10}+\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{8\left(\sqrt{10}+\sqrt{2}\right)}{\left(\sqrt{10}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Square \sqrt{10}. Square \sqrt{2}.

$$\frac{8\left(\sqrt{10}+\sqrt{2}\right)}{10-2}$$

Subtract 2 from 10 to get 8.

$$\frac{8\left(\sqrt{10}+\sqrt{2}\right)}{8}$$

Cancel out 8 and 8.

$$\sqrt{10}+\sqrt{2}$$