Solve frac{2(sqrt{6}+sqrt{2})}{sqrt{2}}

$$\frac { 2 ( \sqrt { 6 } + \sqrt { 2 } ) } { \sqrt { 2 } }$$

Evaluate

$$2\left(\sqrt{3}+1\right)\approx 5.464101615$$

Solution Steps

Rationalize the denominator of \frac{2\left(\sqrt{6}+\sqrt{2}\right)}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

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

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

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

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

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}.

$$\frac{2\sqrt{2}\sqrt{3}\sqrt{2}+2\left(\sqrt{2}\right)^{2}}{2}$$

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

$$\frac{2\times 2\sqrt{3}+2\left(\sqrt{2}\right)^{2}}{2}$$

Multiply 2 and 2 to get 4.

$$\frac{4\sqrt{3}+2\left(\sqrt{2}\right)^{2}}{2}$$

The square of \sqrt{2} is 2.

$$\frac{4\sqrt{3}+2\times 2}{2}$$

Multiply 2 and 2 to get 4.

$$\frac{4\sqrt{3}+4}{2}$$