Solve sqrt{(3sqrt{2})^2-(9/sqrt{5})^2}

$$\sqrt { ( 3 \sqrt { 2 } ) ^ { 2 } - ( \frac { 9 } { \sqrt { 5 } } ) ^ { 2 } }$$

Evaluate

$$\frac{3\sqrt{5}}{5}\approx 1.341640786$$

Solution Steps

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

$$\sqrt{3^{2}\left(\sqrt{2}\right)^{2}-\left(\frac{9}{\sqrt{5}}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\sqrt{9\left(\sqrt{2}\right)^{2}-\left(\frac{9}{\sqrt{5}}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\sqrt{9\times 2-\left(\frac{9}{\sqrt{5}}\right)^{2}}$$

Multiply 9 and 2 to get 18.

$$\sqrt{18-\left(\frac{9}{\sqrt{5}}\right)^{2}}$$

Rationalize the denominator of \frac{9}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\sqrt{18-\left(\frac{9\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\sqrt{18-\left(\frac{9\sqrt{5}}{5}\right)^{2}}$$

To raise \frac{9\sqrt{5}}{5} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{18-\frac{\left(9\sqrt{5}\right)^{2}}{5^{2}}}$$

Expand \left(9\sqrt{5}\right)^{2}.

$$\sqrt{18-\frac{9^{2}\left(\sqrt{5}\right)^{2}}{5^{2}}}$$

Calculate 9 to the power of 2 and get 81.

$$\sqrt{18-\frac{81\left(\sqrt{5}\right)^{2}}{5^{2}}}$$

The square of \sqrt{5} is 5.

$$\sqrt{18-\frac{81\times 5}{5^{2}}}$$

Multiply 81 and 5 to get 405.

$$\sqrt{18-\frac{405}{5^{2}}}$$

Calculate 5 to the power of 2 and get 25.

$$\sqrt{18-\frac{405}{25}}$$

Reduce the fraction \frac{405}{25} to lowest terms by extracting and canceling out 5.

$$\sqrt{18-\frac{81}{5}}$$

Subtract \frac{81}{5} from 18 to get \frac{9}{5}.

$$\sqrt{\frac{9}{5}}$$

Rewrite the square root of the division \sqrt{\frac{9}{5}} as the division of square roots \frac{\sqrt{9}}{\sqrt{5}}.

$$\frac{\sqrt{9}}{\sqrt{5}}$$

Calculate the square root of 9 and get 3.

$$\frac{3}{\sqrt{5}}$$

Rationalize the denominator of \frac{3}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{3\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{3\sqrt{5}}{5}$$