Solve sqrt{(frac{sqrt{65}}{2})^2-(frac{sqrt{15}}{2})^2}

$$\sqrt { ( \frac { \sqrt { 65 } } { 2 } ) ^ { 2 } - ( \frac { \sqrt { 15 } } { 2 } ) ^ { 2 } }$$

Evaluate

$$\frac{5\sqrt{2}}{2}\approx 3.535533906$$

Solution Steps

To raise \frac{\sqrt{65}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{\frac{\left(\sqrt{65}\right)^{2}}{2^{2}}-\left(\frac{\sqrt{15}}{2}\right)^{2}}$$

To raise \frac{\sqrt{15}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{\frac{\left(\sqrt{65}\right)^{2}}{2^{2}}-\frac{\left(\sqrt{15}\right)^{2}}{2^{2}}}$$

The square of \sqrt{15} is 15.

$$\sqrt{\frac{\left(\sqrt{65}\right)^{2}}{2^{2}}-\frac{15}{2^{2}}}$$

Calculate 2 to the power of 2 and get 4.

$$\sqrt{\frac{\left(\sqrt{65}\right)^{2}}{2^{2}}-\frac{15}{4}}$$

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

$$\sqrt{\frac{\left(\sqrt{65}\right)^{2}}{4}-\frac{15}{4}}$$

Since \frac{\left(\sqrt{65}\right)^{2}}{4} and \frac{15}{4} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\left(\sqrt{65}\right)^{2}-15}{4}}$$

The square of \sqrt{65} is 65.

$$\sqrt{\frac{65-15}{4}}$$

Subtract 15 from 65 to get 50.

$$\sqrt{\frac{50}{4}}$$

Reduce the fraction \frac{50}{4} to lowest terms by extracting and canceling out 2.

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

Rewrite the square root of the division \sqrt{\frac{25}{2}} as the division of square roots \frac{\sqrt{25}}{\sqrt{2}}.

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

Calculate the square root of 25 and get 5.

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

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

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

The square of \sqrt{2} is 2.

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