Solve (frac{3sqrt{2}}{5})^2+(frac{5sqrt{2}}{20})^2

$$( \frac { 3 \sqrt { 2 } } { 5 } ) ^ { 2 } + ( \frac { 5 \sqrt { 2 } } { 20 } ) ^ { 2 }$$

Evaluate

$$\frac{169}{200}=0.845$$

Solution Steps

To raise \frac{3\sqrt{2}}{5} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(3\sqrt{2}\right)^{2}}{5^{2}}+\left(\frac{5\sqrt{2}}{20}\right)^{2}$$

Divide 5\sqrt{2} by 20 to get \frac{1}{4}\sqrt{2}.

$$\frac{\left(3\sqrt{2}\right)^{2}}{5^{2}}+\left(\frac{1}{4}\sqrt{2}\right)^{2}$$

Expand \left(\frac{1}{4}\sqrt{2}\right)^{2}.

$$\frac{\left(3\sqrt{2}\right)^{2}}{5^{2}}+\left(\frac{1}{4}\right)^{2}\left(\sqrt{2}\right)^{2}$$

Calculate \frac{1}{4} to the power of 2 and get \frac{1}{16}.

$$\frac{\left(3\sqrt{2}\right)^{2}}{5^{2}}+\frac{1}{16}\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$\frac{\left(3\sqrt{2}\right)^{2}}{5^{2}}+\frac{1}{16}\times 2$$

Multiply \frac{1}{16} and 2 to get \frac{1}{8}.

$$\frac{\left(3\sqrt{2}\right)^{2}}{5^{2}}+\frac{1}{8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5^{2} and 8 is 200. Multiply \frac{\left(3\sqrt{2}\right)^{2}}{5^{2}} times \frac{8}{8}. Multiply \frac{1}{8} times \frac{25}{25}.

$$\frac{8\times \left(3\sqrt{2}\right)^{2}}{200}+\frac{25}{200}$$

Since \frac{8\times \left(3\sqrt{2}\right)^{2}}{200} and \frac{25}{200} have the same denominator, add them by adding their numerators.

$$\frac{8\times \left(3\sqrt{2}\right)^{2}+25}{200}$$

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

$$\frac{3^{2}\left(\sqrt{2}\right)^{2}}{5^{2}}+\frac{1}{8}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{9\left(\sqrt{2}\right)^{2}}{5^{2}}+\frac{1}{8}$$

The square of \sqrt{2} is 2.

$$\frac{9\times 2}{5^{2}}+\frac{1}{8}$$

Multiply 9 and 2 to get 18.

$$\frac{18}{5^{2}}+\frac{1}{8}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{18}{25}+\frac{1}{8}$$

Add \frac{18}{25} and \frac{1}{8} to get \frac{169}{200}.

$$\frac{169}{200}$$

Factor

$$\frac{13 ^ {2}}{2 ^ {3} \cdot 5 ^ {2}} = 0.845$$