Solve sqrt{(2sqrt{89})^2-(frac{sqrt{2}}{2})^2}

$$\sqrt { ( 2 \sqrt { 89 } ) ^ { 2 } - ( \frac { \sqrt { 2 } } { 2 } ) ^ { 2 } }$$

Evaluate

$$\frac{3\sqrt{158}}{2}\approx 18.854707635$$

Solution Steps

Expand \left(2\sqrt{89}\right)^{2}.

$$\sqrt{2^{2}\left(\sqrt{89}\right)^{2}-\left(\frac{\sqrt{2}}{2}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\sqrt{4\left(\sqrt{89}\right)^{2}-\left(\frac{\sqrt{2}}{2}\right)^{2}}$$

The square of \sqrt{89} is 89.

$$\sqrt{4\times 89-\left(\frac{\sqrt{2}}{2}\right)^{2}}$$

Multiply 4 and 89 to get 356.

$$\sqrt{356-\left(\frac{\sqrt{2}}{2}\right)^{2}}$$

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

$$\sqrt{356-\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}}$$

The square of \sqrt{2} is 2.

$$\sqrt{356-\frac{2}{2^{2}}}$$

Calculate 2 to the power of 2 and get 4.

$$\sqrt{356-\frac{2}{4}}$$

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

$$\sqrt{356-\frac{1}{2}}$$

Subtract \frac{1}{2} from 356 to get \frac{711}{2}.

$$\sqrt{\frac{711}{2}}$$

Rewrite the square root of the division \sqrt{\frac{711}{2}} as the division of square roots \frac{\sqrt{711}}{\sqrt{2}}.

$$\frac{\sqrt{711}}{\sqrt{2}}$$

Factor 711=3^{2}\times 79. Rewrite the square root of the product \sqrt{3^{2}\times 79} as the product of square roots \sqrt{3^{2}}\sqrt{79}. Take the square root of 3^{2}.

$$\frac{3\sqrt{79}}{\sqrt{2}}$$

Rationalize the denominator of \frac{3\sqrt{79}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{3\sqrt{79}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{3\sqrt{79}\sqrt{2}}{2}$$

To multiply \sqrt{79} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{3\sqrt{158}}{2}$$