Solve frac{sqrt{30}}{sqrt{40}}

$$\frac { \sqrt { 30 } } { \sqrt { 40 } }$$

Evaluate

$$\frac{\sqrt{3}}{2}\approx 0.866025404$$

Solution Steps

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}.

$$\frac{\sqrt{30}}{2\sqrt{10}}$$

Rationalize the denominator of \frac{\sqrt{30}}{2\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{\sqrt{30}\sqrt{10}}{2\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{\sqrt{30}\sqrt{10}}{2\times 10}$$

Factor 30=10\times 3. Rewrite the square root of the product \sqrt{10\times 3} as the product of square roots \sqrt{10}\sqrt{3}.

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

Multiply \sqrt{10} and \sqrt{10} to get 10.

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

Multiply 2 and 10 to get 20.

$$\frac{10\sqrt{3}}{20}$$

Divide 10\sqrt{3} by 20 to get \frac{1}{2}\sqrt{3}.

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