Solve (sqrt{2})^2+2(frac{2sqrt{3}}{9})^2

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

Evaluate

$$\frac{62}{27}\approx 2.296296296$$

Solution Steps

The square of \sqrt{2} is 2.

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

To raise \frac{2\sqrt{3}}{9} to a power, raise both numerator and denominator to the power and then divide.

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

Express 2\times \frac{\left(2\sqrt{3}\right)^{2}}{9^{2}} as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{9^{2}}{9^{2}}.

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

Since \frac{2\times 9^{2}}{9^{2}} and \frac{2\times \left(2\sqrt{3}\right)^{2}}{9^{2}} have the same denominator, add them by adding their numerators.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

$$2+\frac{2\times 4\times 3}{9^{2}}$$

Multiply 4 and 3 to get 12.

$$2+\frac{2\times 12}{9^{2}}$$

Multiply 2 and 12 to get 24.

$$2+\frac{24}{9^{2}}$$

Calculate 9 to the power of 2 and get 81.

$$2+\frac{24}{81}$$

Reduce the fraction \frac{24}{81} to lowest terms by extracting and canceling out 3.

$$2+\frac{8}{27}$$

Add 2 and \frac{8}{27} to get \frac{62}{27}.

$$\frac{62}{27}$$

Factor

$$\frac{2 \cdot 31}{3 ^ {3}} = 2\frac{8}{27} = 2.2962962962962963$$