Solve sqrt{{left(3sqrt{2}right)}^2-{left(frac{sqrt{30}}{2}right)}^2}

$$\sqrt{ { \left(3 \sqrt{ 2 } \right) }^{ 2 } - { \left( \frac{ \sqrt{ 30 } }{ 2 } \right) }^{ 2 } }$$

Evaluate

$$\frac{\sqrt{42}}{2}\approx 3.240370349$$

Solution Steps

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

$$\sqrt{3^{2}\left(\sqrt{2}\right)^{2}-\left(\frac{\sqrt{30}}{2}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\sqrt{9\left(\sqrt{2}\right)^{2}-\left(\frac{\sqrt{30}}{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\sqrt{9\times 2-\left(\frac{\sqrt{30}}{2}\right)^{2}}$$

Multiply 9 and 2 to get 18.

$$\sqrt{18-\left(\frac{\sqrt{30}}{2}\right)^{2}}$$

To raise \frac{\sqrt{30}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{18-\frac{\left(\sqrt{30}\right)^{2}}{2^{2}}}$$

The square of \sqrt{30} is 30.

$$\sqrt{18-\frac{30}{2^{2}}}$$

Calculate 2 to the power of 2 and get 4.

$$\sqrt{18-\frac{30}{4}}$$

Reduce the fraction \frac{30}{4} to lowest terms by extracting and canceling out 2.

$$\sqrt{18-\frac{15}{2}}$$

Subtract \frac{15}{2} from 18 to get \frac{21}{2}.

$$\sqrt{\frac{21}{2}}$$

Rewrite the square root of the division \sqrt{\frac{21}{2}} as the division of square roots \frac{\sqrt{21}}{\sqrt{2}}.

$$\frac{\sqrt{21}}{\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{21}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{21}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{21}\sqrt{2}}{2}$$

To multiply \sqrt{21} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\sqrt{42}}{2}$$