Solve =sqrt{(frac{13sqrt{2}}{4})^2}+(frac{5sqrt{2}}{4})^2

$$= \sqrt { ( \frac { 13 \sqrt { 2 } } { 4 } ) ^ { 2 } } + ( \frac { 5 \sqrt { 2 } } { 4 } ) ^ { 2 }$$

Evaluate

$$\frac{13\sqrt{2}}{4}+\frac{25}{8}\approx 7.721194078$$

Short Solution Steps

To raise \frac{13\sqrt{2}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{\frac{\left(13\sqrt{2}\right)^{2}}{4^{2}}}+\left(\frac{5\sqrt{2}}{4}\right)^{2}$$

To raise \frac{5\sqrt{2}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{\frac{\left(13\sqrt{2}\right)^{2}}{4^{2}}}+\frac{\left(5\sqrt{2}\right)^{2}}{4^{2}}$$

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

$$\sqrt{\frac{13^{2}\left(\sqrt{2}\right)^{2}}{4^{2}}}+\frac{\left(5\sqrt{2}\right)^{2}}{4^{2}}$$

Calculate 13 to the power of 2 and get 169.

$$\sqrt{\frac{169\left(\sqrt{2}\right)^{2}}{4^{2}}}+\frac{\left(5\sqrt{2}\right)^{2}}{4^{2}}$$

The square of \sqrt{2} is 2.

$$\sqrt{\frac{169\times 2}{4^{2}}}+\frac{\left(5\sqrt{2}\right)^{2}}{4^{2}}$$

Multiply 169 and 2 to get 338.

$$\sqrt{\frac{338}{4^{2}}}+\frac{\left(5\sqrt{2}\right)^{2}}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\sqrt{\frac{338}{16}}+\frac{\left(5\sqrt{2}\right)^{2}}{4^{2}}$$

Reduce the fraction \frac{338}{16} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{169}{8}}+\frac{\left(5\sqrt{2}\right)^{2}}{4^{2}}$$

Rewrite the square root of the division \sqrt{\frac{169}{8}} as the division of square roots \frac{\sqrt{169}}{\sqrt{8}}.

$$\frac{\sqrt{169}}{\sqrt{8}}+\frac{\left(5\sqrt{2}\right)^{2}}{4^{2}}$$

Calculate the square root of 169 and get 13.

$$\frac{13}{\sqrt{8}}+\frac{\left(5\sqrt{2}\right)^{2}}{4^{2}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{13}{2\sqrt{2}}+\frac{\left(5\sqrt{2}\right)^{2}}{4^{2}}$$

Rationalize the denominator of \frac{13}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{13\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}+\frac{\left(5\sqrt{2}\right)^{2}}{4^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{13\sqrt{2}}{2\times 2}+\frac{\left(5\sqrt{2}\right)^{2}}{4^{2}}$$

Multiply 2 and 2 to get 4.

$$\frac{13\sqrt{2}}{4}+\frac{\left(5\sqrt{2}\right)^{2}}{4^{2}}$$

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

$$\frac{13\sqrt{2}}{4}+\frac{5^{2}\left(\sqrt{2}\right)^{2}}{4^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{13\sqrt{2}}{4}+\frac{25\left(\sqrt{2}\right)^{2}}{4^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{13\sqrt{2}}{4}+\frac{25\times 2}{4^{2}}$$

Multiply 25 and 2 to get 50.

$$\frac{13\sqrt{2}}{4}+\frac{50}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{13\sqrt{2}}{4}+\frac{50}{16}$$

Reduce the fraction \frac{50}{16} to lowest terms by extracting and canceling out 2.

$$\frac{13\sqrt{2}}{4}+\frac{25}{8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 8 is 8. Multiply \frac{13\sqrt{2}}{4} times \frac{2}{2}.

$$\frac{2\times 13\sqrt{2}}{8}+\frac{25}{8}$$

Since \frac{2\times 13\sqrt{2}}{8} and \frac{25}{8} have the same denominator, add them by adding their numerators.

$$\frac{2\times 13\sqrt{2}+25}{8}$$

Do the multiplications in 2\times 13\sqrt{2}+25.

$$\frac{26\sqrt{2}+25}{8}$$

Factor

$$\frac{26 \sqrt{2} + 25}{8} = 7.72119407771256$$