Solve frac{(frac{sqrt{6}}{3})^2}{(sqrt{3})^2}+frac{(frac{sqrt{3}}{3})^2}{x^2}=

$$\frac { ( \frac { \sqrt { 6 } } { 3 } ) ^ { 2 } } { ( \sqrt { 3 } ) ^ { 2 } } + \frac { ( \frac { \sqrt { 3 } } { 3 } ) ^ { 2 } } { x ^ { 2 } } =$$

Evaluate

$$\frac{2}{9}+\frac{1}{3x^{2}}$$

Solution Steps

To raise \frac{\sqrt{6}}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(\sqrt{6}\right)^{2}}{3^{2}}}{\left(\sqrt{3}\right)^{2}}+\frac{\left(\frac{\sqrt{3}}{3}\right)^{2}}{x^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{\left(\sqrt{6}\right)^{2}}{3^{2}}}{3}+\frac{\left(\frac{\sqrt{3}}{3}\right)^{2}}{x^{2}}$$

Express \frac{\frac{\left(\sqrt{6}\right)^{2}}{3^{2}}}{3} as a single fraction.

$$\frac{\left(\sqrt{6}\right)^{2}}{3^{2}\times 3}+\frac{\left(\frac{\sqrt{3}}{3}\right)^{2}}{x^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{6}{3^{2}\times 3}+\frac{\left(\frac{\sqrt{3}}{3}\right)^{2}}{x^{2}}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{6}{3^{3}}+\frac{\left(\frac{\sqrt{3}}{3}\right)^{2}}{x^{2}}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{6}{27}+\frac{\left(\frac{\sqrt{3}}{3}\right)^{2}}{x^{2}}$$

Reduce the fraction \frac{6}{27} to lowest terms by extracting and canceling out 3.

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

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

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

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

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

The square of \sqrt{3} is 3.

$$\frac{2}{9}+\frac{3}{3^{2}x^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{2}{9}+\frac{3}{9x^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9 and 9x^{2} is 9x^{2}. Multiply \frac{2}{9} times \frac{x^{2}}{x^{2}}.

$$\frac{2x^{2}}{9x^{2}}+\frac{3}{9x^{2}}$$

Since \frac{2x^{2}}{9x^{2}} and \frac{3}{9x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{2x^{2}+3}{9x^{2}}$$

Factor

$$\frac{2x^{2}+3}{9x^{2}}$$