Solve sqrt{(-frac{sqrt{5}}{5})^2+(frac{2sqrt{5}}{5})^2}

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

Evaluate

$$1$$

Solution Steps

Calculate -\frac{\sqrt{5}}{5} to the power of 2 and get \left(\frac{\sqrt{5}}{5}\right)^{2}.

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

To raise \frac{2\sqrt{5}}{5} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{5} is 5.

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

Multiply 4 and 5 to get 20.

$$\sqrt{\frac{\left(\sqrt{5}\right)^{2}}{5^{2}}+\frac{20}{5^{2}}}$$

Calculate 5 to the power of 2 and get 25.

$$\sqrt{\frac{\left(\sqrt{5}\right)^{2}}{5^{2}}+\frac{20}{25}}$$

Reduce the fraction \frac{20}{25} to lowest terms by extracting and canceling out 5.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5^{2} and 5 is 25. Multiply \frac{4}{5} times \frac{5}{5}.

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

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

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

The square of \sqrt{5} is 5.

$$\sqrt{\frac{5+4\times 5}{25}}$$

Multiply 4 and 5 to get 20.

$$\sqrt{\frac{5+20}{25}}$$

Add 5 and 20 to get 25.

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

Divide 25 by 25 to get 1.

$$\sqrt{1}$$

Calculate the square root of 1 and get 1.

$$1$$

Factor

$$1$$