Solve sqrt{{left(405/82-5/2right)}^2+{left(2/41+3right)}^2}

$$\sqrt{ { \left( \frac{ 405 }{ 82 } - \frac{ 5 }{ 2 } \right) }^{ 2 } + { \left( \frac{ 2 }{ 41 } +3 \right) }^{ 2 } }$$

Evaluate

$$\frac{25\sqrt{41}}{41}\approx 3.904344047$$

Solution Steps

Subtract \frac{5}{2} from \frac{405}{82} to get \frac{100}{41}.

$$\sqrt{\left(\frac{100}{41}\right)^{2}+\left(\frac{2}{41}+3\right)^{2}}$$

Calculate \frac{100}{41} to the power of 2 and get \frac{10000}{1681}.

$$\sqrt{\frac{10000}{1681}+\left(\frac{2}{41}+3\right)^{2}}$$

Add \frac{2}{41} and 3 to get \frac{125}{41}.

$$\sqrt{\frac{10000}{1681}+\left(\frac{125}{41}\right)^{2}}$$

Calculate \frac{125}{41} to the power of 2 and get \frac{15625}{1681}.

$$\sqrt{\frac{10000}{1681}+\frac{15625}{1681}}$$

Add \frac{10000}{1681} and \frac{15625}{1681} to get \frac{625}{41}.

$$\sqrt{\frac{625}{41}}$$

Rewrite the square root of the division \sqrt{\frac{625}{41}} as the division of square roots \frac{\sqrt{625}}{\sqrt{41}}.

$$\frac{\sqrt{625}}{\sqrt{41}}$$

Calculate the square root of 625 and get 25.

$$\frac{25}{\sqrt{41}}$$

Rationalize the denominator of \frac{25}{\sqrt{41}} by multiplying numerator and denominator by \sqrt{41}.

$$\frac{25\sqrt{41}}{\left(\sqrt{41}\right)^{2}}$$

The square of \sqrt{41} is 41.

$$\frac{25\sqrt{41}}{41}$$