Solve sqrt{4^2-(frac{6sqrt{5}}{5})^2}

$$\sqrt { 4 ^ { 2 } - ( \frac { 6 \sqrt { 5 } } { 5 } ) ^ { 2 } }$$

Evaluate

$$\frac{2\sqrt{55}}{5}\approx 2.966479395$$

Solution Steps

Calculate 4 to the power of 2 and get 16.

$$\sqrt{16-\left(\frac{6\sqrt{5}}{5}\right)^{2}}$$

To raise \frac{6\sqrt{5}}{5} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{16-\frac{\left(6\sqrt{5}\right)^{2}}{5^{2}}}$$

Expand \left(6\sqrt{5}\right)^{2}.

$$\sqrt{16-\frac{6^{2}\left(\sqrt{5}\right)^{2}}{5^{2}}}$$

Calculate 6 to the power of 2 and get 36.

$$\sqrt{16-\frac{36\left(\sqrt{5}\right)^{2}}{5^{2}}}$$

The square of \sqrt{5} is 5.

$$\sqrt{16-\frac{36\times 5}{5^{2}}}$$

Multiply 36 and 5 to get 180.

$$\sqrt{16-\frac{180}{5^{2}}}$$

Calculate 5 to the power of 2 and get 25.

$$\sqrt{16-\frac{180}{25}}$$

Reduce the fraction \frac{180}{25} to lowest terms by extracting and canceling out 5.

$$\sqrt{16-\frac{36}{5}}$$

Subtract \frac{36}{5} from 16 to get \frac{44}{5}.

$$\sqrt{\frac{44}{5}}$$

Rewrite the square root of the division \sqrt{\frac{44}{5}} as the division of square roots \frac{\sqrt{44}}{\sqrt{5}}.

$$\frac{\sqrt{44}}{\sqrt{5}}$$

Factor 44=2^{2}\times 11. Rewrite the square root of the product \sqrt{2^{2}\times 11} as the product of square roots \sqrt{2^{2}}\sqrt{11}. Take the square root of 2^{2}.

$$\frac{2\sqrt{11}}{\sqrt{5}}$$

Rationalize the denominator of \frac{2\sqrt{11}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{2\sqrt{11}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{2\sqrt{11}\sqrt{5}}{5}$$

To multiply \sqrt{11} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{2\sqrt{55}}{5}$$