Solve 1/frac{sqrt{3/2}+1/sqrt{3/3}}{sqrt{frac{3}{2}}}

$$\frac{ \frac{ 1 }{ \frac{ \sqrt{ 3 } }{ 2 } } + \frac{ 1 }{ \frac{ \sqrt{ 3 } }{ 3 } } }{ \sqrt{ \frac{ 3 }{ 2 } } }$$

Evaluate

$$\frac{5\sqrt{2}}{3}\approx 2.357022604$$

Solution Steps

Divide 1 by \frac{\sqrt{3}}{2} by multiplying 1 by the reciprocal of \frac{\sqrt{3}}{2}.

$$\frac{\frac{2}{\sqrt{3}}+\frac{1}{\frac{\sqrt{3}}{3}}}{\sqrt{\frac{3}{2}}}$$

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

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

The square of \sqrt{3} is 3.

$$\frac{\frac{2\sqrt{3}}{3}+\frac{1}{\frac{\sqrt{3}}{3}}}{\sqrt{\frac{3}{2}}}$$

Divide 1 by \frac{\sqrt{3}}{3} by multiplying 1 by the reciprocal of \frac{\sqrt{3}}{3}.

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

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

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

The square of \sqrt{3} is 3.

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

Cancel out 3 and 3.

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

Combine \frac{2\sqrt{3}}{3} and \sqrt{3} to get \frac{5}{3}\sqrt{3}.

$$\frac{\frac{5}{3}\sqrt{3}}{\sqrt{\frac{3}{2}}}$$

Rewrite the square root of the division \sqrt{\frac{3}{2}} as the division of square roots \frac{\sqrt{3}}{\sqrt{2}}.

$$\frac{\frac{5}{3}\sqrt{3}}{\frac{\sqrt{3}}{\sqrt{2}}}$$

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

$$\frac{\frac{5}{3}\sqrt{3}}{\frac{\sqrt{3}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{5}{3}\sqrt{3}}{\frac{\sqrt{3}\sqrt{2}}{2}}$$

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

$$\frac{\frac{5}{3}\sqrt{3}}{\frac{\sqrt{6}}{2}}$$

Divide \frac{5}{3}\sqrt{3} by \frac{\sqrt{6}}{2} by multiplying \frac{5}{3}\sqrt{3} by the reciprocal of \frac{\sqrt{6}}{2}.

$$\frac{\frac{5}{3}\sqrt{3}\times 2}{\sqrt{6}}$$

Rationalize the denominator of \frac{\frac{5}{3}\sqrt{3}\times 2}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\frac{5}{3}\sqrt{3}\times 2\sqrt{6}}{\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\frac{5}{3}\sqrt{3}\times 2\sqrt{6}}{6}$$

Express \frac{5}{3}\times 2 as a single fraction.

$$\frac{\frac{5\times 2}{3}\sqrt{3}\sqrt{6}}{6}$$

Multiply 5 and 2 to get 10.

$$\frac{\frac{10}{3}\sqrt{3}\sqrt{6}}{6}$$

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}.

$$\frac{\frac{10}{3}\sqrt{3}\sqrt{3}\sqrt{2}}{6}$$

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

$$\frac{\frac{10}{3}\times 3\sqrt{2}}{6}$$

Cancel out 3 and 3.

$$\frac{10\sqrt{2}}{6}$$

Divide 10\sqrt{2} by 6 to get \frac{5}{3}\sqrt{2}.

$$\frac{5}{3}\sqrt{2}$$