Solve 2sqrt{3}*(1/sqrt{6}+frac{sqrt{3}}{sqrt{2}}

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

Evaluate

$$4\sqrt{2}\approx 5.656854249$$

Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

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

The square of \sqrt{6} is 6.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

Combine \frac{\sqrt{6}}{6} and \frac{\sqrt{6}}{2} to get \frac{2}{3}\sqrt{6}.

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

Express 2\times \frac{2}{3} as a single fraction.

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

Multiply 2 and 2 to get 4.

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

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}.

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

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

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

Cancel out 3 and 3.

$$4\sqrt{2}$$