Solve sqrt{1-(frac{sqrt{3}}{sqrt{7}})^2}

$$\sqrt { 1 - ( \frac { \sqrt { 3 } } { \sqrt { 7 } } ) ^ { 2 } }$$

Evaluate

$$\frac{2\sqrt{7}}{7}\approx 0.755928946$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{3}}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\sqrt{1-\left(\frac{\sqrt{3}\sqrt{7}}{\left(\sqrt{7}\right)^{2}}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\sqrt{1-\left(\frac{\sqrt{3}\sqrt{7}}{7}\right)^{2}}$$

To multiply \sqrt{3} and \sqrt{7}, multiply the numbers under the square root.

$$\sqrt{1-\left(\frac{\sqrt{21}}{7}\right)^{2}}$$

To raise \frac{\sqrt{21}}{7} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{1-\frac{\left(\sqrt{21}\right)^{2}}{7^{2}}}$$

The square of \sqrt{21} is 21.

$$\sqrt{1-\frac{21}{7^{2}}}$$

Calculate 7 to the power of 2 and get 49.

$$\sqrt{1-\frac{21}{49}}$$

Reduce the fraction \frac{21}{49} to lowest terms by extracting and canceling out 7.

$$\sqrt{1-\frac{3}{7}}$$

Subtract \frac{3}{7} from 1 to get \frac{4}{7}.

$$\sqrt{\frac{4}{7}}$$

Rewrite the square root of the division \sqrt{\frac{4}{7}} as the division of square roots \frac{\sqrt{4}}{\sqrt{7}}.

$$\frac{\sqrt{4}}{\sqrt{7}}$$

Calculate the square root of 4 and get 2.

$$\frac{2}{\sqrt{7}}$$

Rationalize the denominator of \frac{2}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{2\sqrt{7}}{\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{2\sqrt{7}}{7}$$