Solve 20/sqrt{6-sqrt{2}}

$$\frac{ 20 }{ \sqrt{ 6 } - \sqrt{ 2 } }$$

Evaluate

$$5\left(\sqrt{2}+\sqrt{6}\right)\approx 19.318516526$$

Solution Steps

Rationalize the denominator of \frac{20}{\sqrt{6}-\sqrt{2}} by multiplying numerator and denominator by \sqrt{6}+\sqrt{2}.

$$\frac{20\left(\sqrt{6}+\sqrt{2}\right)}{\left(\sqrt{6}-\sqrt{2}\right)\left(\sqrt{6}+\sqrt{2}\right)}$$

Consider \left(\sqrt{6}-\sqrt{2}\right)\left(\sqrt{6}+\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}.

$$\frac{20\left(\sqrt{6}+\sqrt{2}\right)}{\left(\sqrt{6}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Square \sqrt{6}. Square \sqrt{2}.

$$\frac{20\left(\sqrt{6}+\sqrt{2}\right)}{6-2}$$

Subtract 2 from 6 to get 4.

$$\frac{20\left(\sqrt{6}+\sqrt{2}\right)}{4}$$

Divide 20\left(\sqrt{6}+\sqrt{2}\right) by 4 to get 5\left(\sqrt{6}+\sqrt{2}\right).

$$5\left(\sqrt{6}+\sqrt{2}\right)$$

Use the distributive property to multiply 5 by \sqrt{6}+\sqrt{2}.

$$5\sqrt{6}+5\sqrt{2}$$