Solve frac{sqrt{2}}{sqrt{5}}frac{sqrt{3}}{sqrt{5}}

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

Evaluate

$$\frac{\sqrt{6}}{5}\approx 0.489897949$$

Solution Steps

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

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

The square of \sqrt{5} is 5.

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

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\sqrt{10}}{5}\times \frac{\sqrt{3}}{\sqrt{5}}$$

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

$$\frac{\sqrt{10}}{5}\times \frac{\sqrt{3}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{10}}{5}\times \frac{\sqrt{3}\sqrt{5}}{5}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\sqrt{10}}{5}\times \frac{\sqrt{15}}{5}$$

Multiply \frac{\sqrt{10}}{5} times \frac{\sqrt{15}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{10}\sqrt{15}}{5\times 5}$$

To multiply \sqrt{10} and \sqrt{15}, multiply the numbers under the square root.

$$\frac{\sqrt{150}}{5\times 5}$$

Multiply 5 and 5 to get 25.

$$\frac{\sqrt{150}}{25}$$

Factor 150=5^{2}\times 6. Rewrite the square root of the product \sqrt{5^{2}\times 6} as the product of square roots \sqrt{5^{2}}\sqrt{6}. Take the square root of 5^{2}.

$$\frac{5\sqrt{6}}{25}$$

Divide 5\sqrt{6} by 25 to get \frac{1}{5}\sqrt{6}.

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