Solve sqrt{1-(frac{3sqrt{7}}{14})^2}

$$\sqrt { 1 - ( \frac { 3 \sqrt { 7 } } { 14 } ) ^ { 2 } }$$

Evaluate

$$\frac{\sqrt{133}}{14}\approx 0.823754471$$

Solution Steps

To raise \frac{3\sqrt{7}}{14} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{1-\frac{\left(3\sqrt{7}\right)^{2}}{14^{2}}}$$

Expand \left(3\sqrt{7}\right)^{2}.

$$\sqrt{1-\frac{3^{2}\left(\sqrt{7}\right)^{2}}{14^{2}}}$$

Calculate 3 to the power of 2 and get 9.

$$\sqrt{1-\frac{9\left(\sqrt{7}\right)^{2}}{14^{2}}}$$

The square of \sqrt{7} is 7.

$$\sqrt{1-\frac{9\times 7}{14^{2}}}$$

Multiply 9 and 7 to get 63.

$$\sqrt{1-\frac{63}{14^{2}}}$$

Calculate 14 to the power of 2 and get 196.

$$\sqrt{1-\frac{63}{196}}$$

Reduce the fraction \frac{63}{196} to lowest terms by extracting and canceling out 7.

$$\sqrt{1-\frac{9}{28}}$$

Subtract \frac{9}{28} from 1 to get \frac{19}{28}.

$$\sqrt{\frac{19}{28}}$$

Rewrite the square root of the division \sqrt{\frac{19}{28}} as the division of square roots \frac{\sqrt{19}}{\sqrt{28}}.

$$\frac{\sqrt{19}}{\sqrt{28}}$$

Factor 28=2^{2}\times 7. Rewrite the square root of the product \sqrt{2^{2}\times 7} as the product of square roots \sqrt{2^{2}}\sqrt{7}. Take the square root of 2^{2}.

$$\frac{\sqrt{19}}{2\sqrt{7}}$$

Rationalize the denominator of \frac{\sqrt{19}}{2\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{\sqrt{19}\sqrt{7}}{2\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{\sqrt{19}\sqrt{7}}{2\times 7}$$

To multiply \sqrt{19} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{\sqrt{133}}{2\times 7}$$

Multiply 2 and 7 to get 14.

$$\frac{\sqrt{133}}{14}$$