Solve sqrt{(3sqrt{2})^2-(frac{sqrt{3}}{2})^2}

$$\sqrt { ( 3 \sqrt { 2 } ) ^ { 2 } - ( \frac { \sqrt { 3 } } { 2 } ) ^ { 2 } }$$

Evaluate

$$\frac{\sqrt{69}}{2}\approx 4.153311931$$

Solution Steps

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

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

Calculate 3 to the power of 2 and get 9.

$$\sqrt{9\left(\sqrt{2}\right)^{2}-\left(\frac{\sqrt{3}}{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\sqrt{9\times 2-\left(\frac{\sqrt{3}}{2}\right)^{2}}$$

Multiply 9 and 2 to get 18.

$$\sqrt{18-\left(\frac{\sqrt{3}}{2}\right)^{2}}$$

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

$$\sqrt{18-\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}}$$

The square of \sqrt{3} is 3.

$$\sqrt{18-\frac{3}{2^{2}}}$$

Calculate 2 to the power of 2 and get 4.

$$\sqrt{18-\frac{3}{4}}$$

Subtract \frac{3}{4} from 18 to get \frac{69}{4}.

$$\sqrt{\frac{69}{4}}$$

Rewrite the square root of the division \sqrt{\frac{69}{4}} as the division of square roots \frac{\sqrt{69}}{\sqrt{4}}.

$$\frac{\sqrt{69}}{\sqrt{4}}$$

Calculate the square root of 4 and get 2.

$$\frac{\sqrt{69}}{2}$$