Solve 3/sqrt{6}*frac{sqrt{2}}{3sqrt{40}}

$$\frac { 3 } { \sqrt { 6 } } \cdot \frac { \sqrt { 2 } } { 3 \sqrt { 40 } }$$

Evaluate

$$\frac{\sqrt{30}}{60}\approx 0.091287093$$

Solution Steps

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

$$\frac{3\sqrt{6}}{\left(\sqrt{6}\right)^{2}}\times \frac{\sqrt{2}}{3\sqrt{40}}$$

The square of \sqrt{6} is 6.

$$\frac{3\sqrt{6}}{6}\times \frac{\sqrt{2}}{3\sqrt{40}}$$

Divide 3\sqrt{6} by 6 to get \frac{1}{2}\sqrt{6}.

$$\frac{1}{2}\sqrt{6}\times \frac{\sqrt{2}}{3\sqrt{40}}$$

Factor 40=2^{2}\times 10. Rewrite the square root of the product \sqrt{2^{2}\times 10} as the product of square roots \sqrt{2^{2}}\sqrt{10}. Take the square root of 2^{2}.

$$\frac{1}{2}\sqrt{6}\times \frac{\sqrt{2}}{3\times 2\sqrt{10}}$$

Multiply 3 and 2 to get 6.

$$\frac{1}{2}\sqrt{6}\times \frac{\sqrt{2}}{6\sqrt{10}}$$

Rationalize the denominator of \frac{\sqrt{2}}{6\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{1}{2}\sqrt{6}\times \frac{\sqrt{2}\sqrt{10}}{6\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{1}{2}\sqrt{6}\times \frac{\sqrt{2}\sqrt{10}}{6\times 10}$$

Factor 10=2\times 5. Rewrite the square root of the product \sqrt{2\times 5} as the product of square roots \sqrt{2}\sqrt{5}.

$$\frac{1}{2}\sqrt{6}\times \frac{\sqrt{2}\sqrt{2}\sqrt{5}}{6\times 10}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{1}{2}\sqrt{6}\times \frac{2\sqrt{5}}{6\times 10}$$

Multiply 6 and 10 to get 60.

$$\frac{1}{2}\sqrt{6}\times \frac{2\sqrt{5}}{60}$$

Divide 2\sqrt{5} by 60 to get \frac{1}{30}\sqrt{5}.

$$\frac{1}{2}\sqrt{6}\times \frac{1}{30}\sqrt{5}$$

Multiply \frac{1}{2} times \frac{1}{30} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 1}{2\times 30}\sqrt{6}\sqrt{5}$$

Do the multiplications in the fraction \frac{1\times 1}{2\times 30}.

$$\frac{1}{60}\sqrt{6}\sqrt{5}$$

To multiply \sqrt{6} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{1}{60}\sqrt{30}$$