Solve (frac{4sqrt{13}}{13})^2+(frac{2sqrt{13}}{13}+1sqrt{13})^2

$$( \frac { 4 \sqrt { 13 } } { 13 } ) ^ { 2 } + ( \frac { 2 \sqrt { 13 } } { 13 } + 1 \sqrt { 13 } ) ^ { 2 }$$

Evaluate

$$\frac{241}{13}\approx 18.538461538$$

Solution Steps

To raise \frac{4\sqrt{13}}{13} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(4\sqrt{13}\right)^{2}}{13^{2}}+\left(\frac{2\sqrt{13}}{13}+1\sqrt{13}\right)^{2}$$

Combine \frac{2\sqrt{13}}{13} and 1\sqrt{13} to get \frac{15}{13}\sqrt{13}.

$$\frac{\left(4\sqrt{13}\right)^{2}}{13^{2}}+\left(\frac{15}{13}\sqrt{13}\right)^{2}$$

Expand \left(\frac{15}{13}\sqrt{13}\right)^{2}.

$$\frac{\left(4\sqrt{13}\right)^{2}}{13^{2}}+\left(\frac{15}{13}\right)^{2}\left(\sqrt{13}\right)^{2}$$

Calculate \frac{15}{13} to the power of 2 and get \frac{225}{169}.

$$\frac{\left(4\sqrt{13}\right)^{2}}{13^{2}}+\frac{225}{169}\left(\sqrt{13}\right)^{2}$$

The square of \sqrt{13} is 13.

$$\frac{\left(4\sqrt{13}\right)^{2}}{13^{2}}+\frac{225}{169}\times 13$$

Multiply \frac{225}{169} and 13 to get \frac{225}{13}.

$$\frac{\left(4\sqrt{13}\right)^{2}}{13^{2}}+\frac{225}{13}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 13^{2} and 13 is 169. Multiply \frac{225}{13} times \frac{13}{13}.

$$\frac{\left(4\sqrt{13}\right)^{2}}{169}+\frac{225\times 13}{169}$$

Since \frac{\left(4\sqrt{13}\right)^{2}}{169} and \frac{225\times 13}{169} have the same denominator, add them by adding their numerators.

$$\frac{\left(4\sqrt{13}\right)^{2}+225\times 13}{169}$$

Expand \left(4\sqrt{13}\right)^{2}.

$$\frac{4^{2}\left(\sqrt{13}\right)^{2}}{13^{2}}+\frac{225}{13}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{16\left(\sqrt{13}\right)^{2}}{13^{2}}+\frac{225}{13}$$

The square of \sqrt{13} is 13.

$$\frac{16\times 13}{13^{2}}+\frac{225}{13}$$

Multiply 16 and 13 to get 208.

$$\frac{208}{13^{2}}+\frac{225}{13}$$

Calculate 13 to the power of 2 and get 169.

$$\frac{208}{169}+\frac{225}{13}$$

Reduce the fraction \frac{208}{169} to lowest terms by extracting and canceling out 13.

$$\frac{16}{13}+\frac{225}{13}$$

Add \frac{16}{13} and \frac{225}{13} to get \frac{241}{13}.

$$\frac{241}{13}$$

Factor

$$\frac{241}{13} = 18\frac{7}{13} = 18.53846153846154$$