Solve sqrt{(frac{5sqrt{3}}{2})^2+(15/2)^2}

$$\sqrt { ( \frac { 5 \sqrt { 3 } } { 2 } ) ^ { 2 } + ( \frac { 15 } { 2 } ) ^ { 2 } }$$

Evaluate

$$5\sqrt{3}\approx 8.660254038$$

Solution Steps

To raise \frac{5\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{\frac{\left(5\sqrt{3}\right)^{2}}{2^{2}}+\left(\frac{15}{2}\right)^{2}}$$

Calculate \frac{15}{2} to the power of 2 and get \frac{225}{4}.

$$\sqrt{\frac{\left(5\sqrt{3}\right)^{2}}{2^{2}}+\frac{225}{4}}$$

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

$$\sqrt{\frac{\left(5\sqrt{3}\right)^{2}}{4}+\frac{225}{4}}$$

Since \frac{\left(5\sqrt{3}\right)^{2}}{4} and \frac{225}{4} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\left(5\sqrt{3}\right)^{2}+225}{4}}$$

Expand \left(5\sqrt{3}\right)^{2}.

$$\sqrt{\frac{5^{2}\left(\sqrt{3}\right)^{2}+225}{4}}$$

Calculate 5 to the power of 2 and get 25.

$$\sqrt{\frac{25\left(\sqrt{3}\right)^{2}+225}{4}}$$

The square of \sqrt{3} is 3.

$$\sqrt{\frac{25\times 3+225}{4}}$$

Multiply 25 and 3 to get 75.

$$\sqrt{\frac{75+225}{4}}$$

Add 75 and 225 to get 300.

$$\sqrt{\frac{300}{4}}$$

Divide 300 by 4 to get 75.

$$\sqrt{75}$$

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$5\sqrt{3}$$