Solve sqrt{(frac{10sqrt{39}}{13})^2+(frac{5sqrt{13}}{13}+sqrt{13})^2}

$$\sqrt { ( \frac { 10 \sqrt { 39 } } { 13 } ) ^ { 2 } + ( \frac { 5 \sqrt { 13 } } { 13 } + \sqrt { 13 } ) ^ { 2 } }$$

Evaluate

$$4\sqrt{3}\approx 6.92820323$$

Solution Steps

To raise \frac{10\sqrt{39}}{13} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{\frac{\left(10\sqrt{39}\right)^{2}}{13^{2}}+\left(\frac{5\sqrt{13}}{13}+\sqrt{13}\right)^{2}}$$

Combine \frac{5\sqrt{13}}{13} and \sqrt{13} to get \frac{18}{13}\sqrt{13}.

$$\sqrt{\frac{\left(10\sqrt{39}\right)^{2}}{13^{2}}+\left(\frac{18}{13}\sqrt{13}\right)^{2}}$$

Expand \left(\frac{18}{13}\sqrt{13}\right)^{2}.

$$\sqrt{\frac{\left(10\sqrt{39}\right)^{2}}{13^{2}}+\left(\frac{18}{13}\right)^{2}\left(\sqrt{13}\right)^{2}}$$

Calculate \frac{18}{13} to the power of 2 and get \frac{324}{169}.

$$\sqrt{\frac{\left(10\sqrt{39}\right)^{2}}{13^{2}}+\frac{324}{169}\left(\sqrt{13}\right)^{2}}$$

The square of \sqrt{13} is 13.

$$\sqrt{\frac{\left(10\sqrt{39}\right)^{2}}{13^{2}}+\frac{324}{169}\times 13}$$

Multiply \frac{324}{169} and 13 to get \frac{324}{13}.

$$\sqrt{\frac{\left(10\sqrt{39}\right)^{2}}{13^{2}}+\frac{324}{13}}$$

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{324}{13} times \frac{13}{13}.

$$\sqrt{\frac{\left(10\sqrt{39}\right)^{2}}{169}+\frac{324\times 13}{169}}$$

Since \frac{\left(10\sqrt{39}\right)^{2}}{169} and \frac{324\times 13}{169} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\left(10\sqrt{39}\right)^{2}+324\times 13}{169}}$$

Expand \left(10\sqrt{39}\right)^{2}.

$$\sqrt{\frac{10^{2}\left(\sqrt{39}\right)^{2}+324\times 13}{169}}$$

Calculate 10 to the power of 2 and get 100.

$$\sqrt{\frac{100\left(\sqrt{39}\right)^{2}+324\times 13}{169}}$$

The square of \sqrt{39} is 39.

$$\sqrt{\frac{100\times 39+324\times 13}{169}}$$

Multiply 100 and 39 to get 3900.

$$\sqrt{\frac{3900+324\times 13}{169}}$$

Multiply 324 and 13 to get 4212.

$$\sqrt{\frac{3900+4212}{169}}$$

Add 3900 and 4212 to get 8112.

$$\sqrt{\frac{8112}{169}}$$

Divide 8112 by 169 to get 48.

$$\sqrt{48}$$

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$4\sqrt{3}$$