Solve sqrt{(frac{4sqrt{5}}{5})^2+25}

$$\sqrt { ( \frac { 4 \sqrt { 5 } } { 5 } ) ^ { 2 } + 25 }$$

Evaluate

$$\frac{\sqrt{705}}{5}\approx 5.310367219$$

Solution Steps

To raise \frac{4\sqrt{5}}{5} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{\frac{\left(4\sqrt{5}\right)^{2}}{5^{2}}+25}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 25 times \frac{5^{2}}{5^{2}}.

$$\sqrt{\frac{\left(4\sqrt{5}\right)^{2}}{5^{2}}+\frac{25\times 5^{2}}{5^{2}}}$$

Since \frac{\left(4\sqrt{5}\right)^{2}}{5^{2}} and \frac{25\times 5^{2}}{5^{2}} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\left(4\sqrt{5}\right)^{2}+25\times 5^{2}}{5^{2}}}$$

Expand \left(4\sqrt{5}\right)^{2}.

$$\sqrt{\frac{4^{2}\left(\sqrt{5}\right)^{2}+25\times 5^{2}}{5^{2}}}$$

Calculate 4 to the power of 2 and get 16.

$$\sqrt{\frac{16\left(\sqrt{5}\right)^{2}+25\times 5^{2}}{5^{2}}}$$

The square of \sqrt{5} is 5.

$$\sqrt{\frac{16\times 5+25\times 5^{2}}{5^{2}}}$$

Multiply 16 and 5 to get 80.

$$\sqrt{\frac{80+25\times 5^{2}}{5^{2}}}$$

Calculate 5 to the power of 2 and get 25.

$$\sqrt{\frac{80+25\times 25}{5^{2}}}$$

Multiply 25 and 25 to get 625.

$$\sqrt{\frac{80+625}{5^{2}}}$$

Add 80 and 625 to get 705.

$$\sqrt{\frac{705}{5^{2}}}$$

Calculate 5 to the power of 2 and get 25.

$$\sqrt{\frac{705}{25}}$$

Reduce the fraction \frac{705}{25} to lowest terms by extracting and canceling out 5.

$$\sqrt{\frac{141}{5}}$$

Rewrite the square root of the division \sqrt{\frac{141}{5}} as the division of square roots \frac{\sqrt{141}}{\sqrt{5}}.

$$\frac{\sqrt{141}}{\sqrt{5}}$$

Rationalize the denominator of \frac{\sqrt{141}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\sqrt{141}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{141}\sqrt{5}}{5}$$

To multiply \sqrt{141} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\sqrt{705}}{5}$$