Solve sqrt{25-{left(frac{sqrt{2}}{2}right)}^2}

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

Evaluate

$$\frac{7\sqrt{2}}{2}\approx 4.949747468$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

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

Calculate 2 to the power of 2 and get 4.

$$\sqrt{25-\frac{2}{4}}$$

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

$$\sqrt{25-\frac{1}{2}}$$

Subtract \frac{1}{2} from 25 to get \frac{49}{2}.

$$\sqrt{\frac{49}{2}}$$

Rewrite the square root of the division \sqrt{\frac{49}{2}} as the division of square roots \frac{\sqrt{49}}{\sqrt{2}}.

$$\frac{\sqrt{49}}{\sqrt{2}}$$

Calculate the square root of 49 and get 7.

$$\frac{7}{\sqrt{2}}$$

Rationalize the denominator of \frac{7}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{7\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{7\sqrt{2}}{2}$$