Solve sqrt{{left(frac{16sqrt{13}}{13}right)}^2+{left(frac{2sqrt{13}}{13}right)}^2}

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

Evaluate

$$2\sqrt{5}\approx 4.472135955$$

Solution Steps

To raise \frac{16\sqrt{13}}{13} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Since \frac{\left(16\sqrt{13}\right)^{2}}{13^{2}} and \frac{\left(2\sqrt{13}\right)^{2}}{13^{2}} have the same denominator, add them by adding their numerators.

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

Expand \left(16\sqrt{13}\right)^{2}.

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

Calculate 16 to the power of 2 and get 256.

$$\sqrt{\frac{256\left(\sqrt{13}\right)^{2}+\left(2\sqrt{13}\right)^{2}}{13^{2}}}$$

The square of \sqrt{13} is 13.

$$\sqrt{\frac{256\times 13+\left(2\sqrt{13}\right)^{2}}{13^{2}}}$$

Multiply 256 and 13 to get 3328.

$$\sqrt{\frac{3328+\left(2\sqrt{13}\right)^{2}}{13^{2}}}$$

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

$$\sqrt{\frac{3328+2^{2}\left(\sqrt{13}\right)^{2}}{13^{2}}}$$

Calculate 2 to the power of 2 and get 4.

$$\sqrt{\frac{3328+4\left(\sqrt{13}\right)^{2}}{13^{2}}}$$

The square of \sqrt{13} is 13.

$$\sqrt{\frac{3328+4\times 13}{13^{2}}}$$

Multiply 4 and 13 to get 52.

$$\sqrt{\frac{3328+52}{13^{2}}}$$

Add 3328 and 52 to get 3380.

$$\sqrt{\frac{3380}{13^{2}}}$$

Calculate 13 to the power of 2 and get 169.

$$\sqrt{\frac{3380}{169}}$$

Divide 3380 by 169 to get 20.

$$\sqrt{20}$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$2\sqrt{5}$$