Add $ \dfrac{x^2}{36} $ and $ \dfrac{1}{3} $ to get $ \dfrac{ \color{purple}{ x^2+12 } }{ 36 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{12}$.

$$ \begin{aligned} \frac{x^2}{36} + \frac{1}{3} & = \frac{ x^2 }{ 36 } + \frac{ 1 \cdot \color{blue}{ 12 }}{ 3 \cdot \color{blue}{ 12 }} = \\[1ex] &=\frac{ \color{purple}{ x^2 } }{ 36 } + \frac{ \color{purple}{ 12 } }{ 36 }=\frac{ \color{purple}{ x^2+12 } }{ 36 } \end{aligned} $$

Subtract $ \dfrac{1}{3} $ from $ \dfrac{x+1}{9} $ to get $ \dfrac{ \color{purple}{ x-2 } }{ 9 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{3}$.

$$ \begin{aligned} \frac{x+1}{9} - \frac{1}{3} & = \frac{ x+1 }{ 9 } - \frac{ 1 \cdot \color{blue}{ 3 }}{ 3 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ x+1 } }{ 9 } - \frac{ \color{purple}{ 3 } }{ 9 }=\frac{ \color{purple}{ x-2 } }{ 9 } \end{aligned} $$

Divide $ \dfrac{x^2+12}{36} $ by $ \dfrac{x-2}{9} $ to get $ \dfrac{ 9x^2+108 }{ 36x-72 } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{x^2+12}{36} }{ \frac{\color{blue}{x-2}}{\color{blue}{9}} } & \xlongequal{\text{Step 1}} \frac{x^2+12}{36} \cdot \frac{\color{blue}{9}}{\color{blue}{x-2}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^2+12 \right) \cdot 9 }{ 36 \cdot \left( x-2 \right) } \xlongequal{\text{Step 3}} \frac{ 9x^2+108 }{ 36x-72 } \end{aligned} $$