Solve sqrt{({left(975/2right)}^2}+{left(frac{375sqrt{3}}{2}right)}^2)

$$\sqrt{ ( { \left( \frac{ 975 }{ 2 } \right) }^{ 2 } } + { \left( \frac{ 375 \sqrt{ 3 } }{ 2 } \right) }^{ 2 } )$$

Evaluate

$$75\sqrt{61}\approx 585.768725693$$

Solution Steps

Calculate \frac{975}{2} to the power of 2 and get \frac{950625}{4}.

$$\sqrt{\frac{950625}{4}+\left(\frac{375\sqrt{3}}{2}\right)^{2}}$$

To raise \frac{375\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{\frac{950625}{4}+\frac{\left(375\sqrt{3}\right)^{2}}{2^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Expand 2^{2}.

$$\sqrt{\frac{950625}{4}+\frac{\left(375\sqrt{3}\right)^{2}}{4}}$$

Since \frac{950625}{4} and \frac{\left(375\sqrt{3}\right)^{2}}{4} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{950625+\left(375\sqrt{3}\right)^{2}}{4}}$$

Expand \left(375\sqrt{3}\right)^{2}.

$$\sqrt{\frac{950625+375^{2}\left(\sqrt{3}\right)^{2}}{4}}$$

Calculate 375 to the power of 2 and get 140625.

$$\sqrt{\frac{950625+140625\left(\sqrt{3}\right)^{2}}{4}}$$

The square of \sqrt{3} is 3.

$$\sqrt{\frac{950625+140625\times 3}{4}}$$

Multiply 140625 and 3 to get 421875.

$$\sqrt{\frac{950625+421875}{4}}$$

Add 950625 and 421875 to get 1372500.

$$\sqrt{\frac{1372500}{4}}$$

Divide 1372500 by 4 to get 343125.

$$\sqrt{343125}$$

Factor 343125=75^{2}\times 61. Rewrite the square root of the product \sqrt{75^{2}\times 61} as the product of square roots \sqrt{75^{2}}\sqrt{61}. Take the square root of 75^{2}.

$$75\sqrt{61}$$