Solve (frac{2sqrt{3}}{3})^2+(frac{4sqrt{3}}{3})^2=

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

Evaluate

$$\frac{20}{3}\approx 6.666666667$$

Solution Steps

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

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

To raise \frac{4\sqrt{3}}{3} to a power, raise both numerator and denominator to the power and then divide.

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

Since \frac{\left(2\sqrt{3}\right)^{2}}{3^{2}} and \frac{\left(4\sqrt{3}\right)^{2}}{3^{2}} have the same denominator, add them by adding their numerators.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

$$\frac{12}{3^{2}}+\frac{\left(4\sqrt{3}\right)^{2}}{3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{12}{9}+\frac{\left(4\sqrt{3}\right)^{2}}{3^{2}}$$

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

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

Expand \left(4\sqrt{3}\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$\frac{4}{3}+\frac{16\left(\sqrt{3}\right)^{2}}{3^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{4}{3}+\frac{16\times 3}{3^{2}}$$

Multiply 16 and 3 to get 48.

$$\frac{4}{3}+\frac{48}{3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{4}{3}+\frac{48}{9}$$

Reduce the fraction \frac{48}{9} to lowest terms by extracting and canceling out 3.

$$\frac{4}{3}+\frac{16}{3}$$

Add \frac{4}{3} and \frac{16}{3} to get \frac{20}{3}.

$$\frac{20}{3}$$

Factor

$$\frac{2 ^ {2} \cdot 5}{3} = 6\frac{2}{3} = 6.666666666666667$$