Solve 8/sqrt{6+sqrt{2}}

$$\frac { 8 } { \sqrt { 6 } + \sqrt { 2 } }$$

Evaluate

$$2\left(\sqrt{6}-\sqrt{2}\right)\approx 2.070552361$$

Solution Steps

Rationalize the denominator of \frac{8}{\sqrt{6}+\sqrt{2}} by multiplying numerator and denominator by \sqrt{6}-\sqrt{2}.

$$\frac{8\left(\sqrt{6}-\sqrt{2}\right)}{\left(\sqrt{6}+\sqrt{2}\right)\left(\sqrt{6}-\sqrt{2}\right)}$$

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}.

$$\frac{8\left(\sqrt{6}-\sqrt{2}\right)}{\left(\sqrt{6}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

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

$$\frac{8\left(\sqrt{6}-\sqrt{2}\right)}{6-2}$$

Subtract 2 from 6 to get 4.

$$\frac{8\left(\sqrt{6}-\sqrt{2}\right)}{4}$$

Divide 8\left(\sqrt{6}-\sqrt{2}\right) by 4 to get 2\left(\sqrt{6}-\sqrt{2}\right).

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

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

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