Solve frac{4sqrt{2}}{sqrt{2}+sqrt{6}}

$$\frac { 4 \sqrt { 2 } } { \sqrt { 2 } + \sqrt { 6 } }$$

Evaluate

$$2\left(\sqrt{3}-1\right)\approx 1.464101615$$

Solution Steps

Rationalize the denominator of \frac{4\sqrt{2}}{\sqrt{2}+\sqrt{6}} by multiplying numerator and denominator by \sqrt{2}-\sqrt{6}.

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

Consider \left(\sqrt{2}+\sqrt{6}\right)\left(\sqrt{2}-\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Subtract 6 from 2 to get -4.

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

Use the distributive property to multiply 4\sqrt{2} by \sqrt{2}-\sqrt{6}.

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

The square of \sqrt{2} is 2.

$$\frac{4\times 2-4\sqrt{2}\sqrt{6}}{-4}$$

Multiply 4 and 2 to get 8.

$$\frac{8-4\sqrt{2}\sqrt{6}}{-4}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$\frac{8-4\sqrt{2}\sqrt{2}\sqrt{3}}{-4}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{8-4\times 2\sqrt{3}}{-4}$$

Multiply -4 and 2 to get -8.

$$\frac{8-8\sqrt{3}}{-4}$$