Solve sqrt{8^2-(frac{28sqrt{19}}{19})^2}

$$\sqrt { 8 ^ { 2 } - ( \frac { 28 \sqrt { 19 } } { 19 } ) ^ { 2 } }$$

Evaluate

$$\frac{12\sqrt{57}}{19}\approx 4.768316485$$

Solution Steps

Calculate 8 to the power of 2 and get 64.

$$\sqrt{64-\left(\frac{28\sqrt{19}}{19}\right)^{2}}$$

To raise \frac{28\sqrt{19}}{19} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{64-\frac{\left(28\sqrt{19}\right)^{2}}{19^{2}}}$$

Expand \left(28\sqrt{19}\right)^{2}.

$$\sqrt{64-\frac{28^{2}\left(\sqrt{19}\right)^{2}}{19^{2}}}$$

Calculate 28 to the power of 2 and get 784.

$$\sqrt{64-\frac{784\left(\sqrt{19}\right)^{2}}{19^{2}}}$$

The square of \sqrt{19} is 19.

$$\sqrt{64-\frac{784\times 19}{19^{2}}}$$

Multiply 784 and 19 to get 14896.

$$\sqrt{64-\frac{14896}{19^{2}}}$$

Calculate 19 to the power of 2 and get 361.

$$\sqrt{64-\frac{14896}{361}}$$

Reduce the fraction \frac{14896}{361} to lowest terms by extracting and canceling out 19.

$$\sqrt{64-\frac{784}{19}}$$

Subtract \frac{784}{19} from 64 to get \frac{432}{19}.

$$\sqrt{\frac{432}{19}}$$

Rewrite the square root of the division \sqrt{\frac{432}{19}} as the division of square roots \frac{\sqrt{432}}{\sqrt{19}}.

$$\frac{\sqrt{432}}{\sqrt{19}}$$

Factor 432=12^{2}\times 3. Rewrite the square root of the product \sqrt{12^{2}\times 3} as the product of square roots \sqrt{12^{2}}\sqrt{3}. Take the square root of 12^{2}.

$$\frac{12\sqrt{3}}{\sqrt{19}}$$

Rationalize the denominator of \frac{12\sqrt{3}}{\sqrt{19}} by multiplying numerator and denominator by \sqrt{19}.

$$\frac{12\sqrt{3}\sqrt{19}}{\left(\sqrt{19}\right)^{2}}$$

The square of \sqrt{19} is 19.

$$\frac{12\sqrt{3}\sqrt{19}}{19}$$

To multiply \sqrt{3} and \sqrt{19}, multiply the numbers under the square root.

$$\frac{12\sqrt{57}}{19}$$