Solve frac{6+2sqrt{3}}{4sqrt{6}}

$$\frac { 6 + 2 \sqrt { 3 } } { 4 \sqrt { 6 } }$$

Evaluate

$$\frac{\sqrt{2}\left(\sqrt{3}+1\right)}{4}\approx 0.965925826$$

Solution Steps

Rationalize the denominator of \frac{6+2\sqrt{3}}{4\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\left(6+2\sqrt{3}\right)\sqrt{6}}{4\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\left(6+2\sqrt{3}\right)\sqrt{6}}{4\times 6}$$

Multiply 4 and 6 to get 24.

$$\frac{\left(6+2\sqrt{3}\right)\sqrt{6}}{24}$$

Use the distributive property to multiply 6+2\sqrt{3} by \sqrt{6}.

$$\frac{6\sqrt{6}+2\sqrt{3}\sqrt{6}}{24}$$

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

$$\frac{6\sqrt{6}+2\sqrt{3}\sqrt{3}\sqrt{2}}{24}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{6\sqrt{6}+2\times 3\sqrt{2}}{24}$$

Multiply 2 and 3 to get 6.

$$\frac{6\sqrt{6}+6\sqrt{2}}{24}$$

Factor

$$\frac{\sqrt{2} {(\sqrt{3} + 1)}}{4} = 0.9659258262890683$$