Solve sqrt{1/3}(2sqrt{3}+6sqrt{2})

$$\sqrt{ \frac{ 1 }{ 3 } } (2 \sqrt{ 3 } +6 \sqrt{ 2 } )$$

Evaluate

$$2\left(\sqrt{6}+1\right)\approx 6.898979486$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{3}} as the division of square roots \frac{\sqrt{1}}{\sqrt{3}}.

$$\frac{\sqrt{1}}{\sqrt{3}}\left(2\sqrt{3}+6\sqrt{2}\right)$$

Calculate the square root of 1 and get 1.

$$\frac{1}{\sqrt{3}}\left(2\sqrt{3}+6\sqrt{2}\right)$$

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

Express \frac{\sqrt{3}}{3}\left(2\sqrt{3}+6\sqrt{2}\right) as a single fraction.

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 2 and 3 to get 6.

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

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

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

Factor

$$2 {(\sqrt{6} + 1)} = 6.898979486$$