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

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

Evaluate

$$\frac{\sqrt{15}}{3}\approx 1.290994449$$

Solution Steps

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

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

Calculate \frac{2}{3} to the power of 2 and get \frac{4}{9}.

$$\sqrt{3-\frac{4}{9}\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\sqrt{3-\frac{4}{9}\times 3}$$

Multiply \frac{4}{9} and 3 to get \frac{4}{3}.

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

Subtract \frac{4}{3} from 3 to get \frac{5}{3}.

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

Rewrite the square root of the division \sqrt{\frac{5}{3}} as the division of square roots \frac{\sqrt{5}}{\sqrt{3}}.

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

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

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

The square of \sqrt{3} is 3.

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

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\sqrt{15}}{3}$$