Solve sqrt{4-(frac{2sqrt{3}}{3})^2}

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

Evaluate

$$\frac{2\sqrt{6}}{3}\approx 1.632993162$$

Solution Steps

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

$$\sqrt{4-\frac{4\times 3}{3^{2}}}$$

Multiply 4 and 3 to get 12.

$$\sqrt{4-\frac{12}{3^{2}}}$$

Calculate 3 to the power of 2 and get 9.

$$\sqrt{4-\frac{12}{9}}$$

Reduce the fraction \frac{12}{9} to lowest terms by extracting and canceling out 3.

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

Subtract \frac{4}{3} from 4 to get \frac{8}{3}.

$$\sqrt{\frac{8}{3}}$$

Rewrite the square root of the division \sqrt{\frac{8}{3}} as the division of square roots \frac{\sqrt{8}}{\sqrt{3}}.

$$\frac{\sqrt{8}}{\sqrt{3}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

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

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

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

The square of \sqrt{3} is 3.

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

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{2\sqrt{6}}{3}$$