Solve sqrt{1-(frac{3sqrt{57}}{38})^2}

$$\sqrt { 1 - ( \frac { 3 \sqrt { 57 } } { 38 } ) ^ { 2 } }$$

Evaluate

$$\frac{7\sqrt{19}}{38}\approx 0.802955069$$

Solution Steps

To raise \frac{3\sqrt{57}}{38} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{1-\frac{\left(3\sqrt{57}\right)^{2}}{38^{2}}}$$

Expand \left(3\sqrt{57}\right)^{2}.

$$\sqrt{1-\frac{3^{2}\left(\sqrt{57}\right)^{2}}{38^{2}}}$$

Calculate 3 to the power of 2 and get 9.

$$\sqrt{1-\frac{9\left(\sqrt{57}\right)^{2}}{38^{2}}}$$

The square of \sqrt{57} is 57.

$$\sqrt{1-\frac{9\times 57}{38^{2}}}$$

Multiply 9 and 57 to get 513.

$$\sqrt{1-\frac{513}{38^{2}}}$$

Calculate 38 to the power of 2 and get 1444.

$$\sqrt{1-\frac{513}{1444}}$$

Reduce the fraction \frac{513}{1444} to lowest terms by extracting and canceling out 19.

$$\sqrt{1-\frac{27}{76}}$$

Subtract \frac{27}{76} from 1 to get \frac{49}{76}.

$$\sqrt{\frac{49}{76}}$$

Rewrite the square root of the division \sqrt{\frac{49}{76}} as the division of square roots \frac{\sqrt{49}}{\sqrt{76}}.

$$\frac{\sqrt{49}}{\sqrt{76}}$$

Calculate the square root of 49 and get 7.

$$\frac{7}{\sqrt{76}}$$

Factor 76=2^{2}\times 19. Rewrite the square root of the product \sqrt{2^{2}\times 19} as the product of square roots \sqrt{2^{2}}\sqrt{19}. Take the square root of 2^{2}.

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

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

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

The square of \sqrt{19} is 19.

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

Multiply 2 and 19 to get 38.

$$\frac{7\sqrt{19}}{38}$$