Solve sqrt{5^2+(5/2sqrt{6})^2}

$$\sqrt { 5 ^ { 2 } + ( \frac { 5 } { 2 } \sqrt { 6 } ) ^ { 2 } }$$

Evaluate

$$\frac{5\sqrt{10}}{2}\approx 7.90569415$$

Solution Steps

Calculate 5 to the power of 2 and get 25.

$$\sqrt{25+\left(\frac{5}{2}\sqrt{6}\right)^{2}}$$

Expand \left(\frac{5}{2}\sqrt{6}\right)^{2}.

$$\sqrt{25+\left(\frac{5}{2}\right)^{2}\left(\sqrt{6}\right)^{2}}$$

Calculate \frac{5}{2} to the power of 2 and get \frac{25}{4}.

$$\sqrt{25+\frac{25}{4}\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\sqrt{25+\frac{25}{4}\times 6}$$

Multiply \frac{25}{4} and 6 to get \frac{75}{2}.

$$\sqrt{25+\frac{75}{2}}$$

Add 25 and \frac{75}{2} to get \frac{125}{2}.

$$\sqrt{\frac{125}{2}}$$

Rewrite the square root of the division \sqrt{\frac{125}{2}} as the division of square roots \frac{\sqrt{125}}{\sqrt{2}}.

$$\frac{\sqrt{125}}{\sqrt{2}}$$

Factor 125=5^{2}\times 5. Rewrite the square root of the product \sqrt{5^{2}\times 5} as the product of square roots \sqrt{5^{2}}\sqrt{5}. Take the square root of 5^{2}.

$$\frac{5\sqrt{5}}{\sqrt{2}}$$

Rationalize the denominator of \frac{5\sqrt{5}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

$$\frac{5\sqrt{5}\sqrt{2}}{2}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{5\sqrt{10}}{2}$$