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

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

Evaluate

$$\frac{\sqrt{6}}{18}\approx 0.136082763$$

Solution Steps

Multiply \frac{1}{2} times \frac{\sqrt{2}}{2} by multiplying numerator times numerator and denominator times denominator.

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

Anything divided by one gives itself.

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

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

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

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

Multiply 2 and 2 to get 4.

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

Express 4\times \frac{3}{2} as a single fraction.

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

Multiply 4 and 3 to get 12.

$$\frac{\sqrt{6}}{\frac{12}{2}\times 3}$$

Divide 12 by 2 to get 6.

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

Multiply 6 and 3 to get 18.

$$\frac{\sqrt{6}}{18}$$