Multiply $ \dfrac{1}{2} $ by $ a $ to get $ \dfrac{ a }{ 2 } $.

Step 1: Write $ a $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{2} \cdot a & \xlongequal{\text{Step 1}} \frac{1}{2} \cdot \frac{a}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot a }{ 2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ a }{ 2 } \end{aligned} $$

Multiply $ \dfrac{2}{3} $ by $ a^2 $ to get $ \dfrac{ 2a^2 }{ 3 } $.

Step 1: Write $ a^2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{2}{3} \cdot a^2 & \xlongequal{\text{Step 1}} \frac{2}{3} \cdot \frac{a^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 2 \cdot a^2 }{ 3 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2a^2 }{ 3 } \end{aligned} $$

Multiply $ \dfrac{3}{4} $ by $ a $ to get $ \dfrac{ 3a }{ 4 } $.

Step 1: Write $ a $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{3}{4} \cdot a & \xlongequal{\text{Step 1}} \frac{3}{4} \cdot \frac{a}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 3 \cdot a }{ 4 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 3a }{ 4 } \end{aligned} $$

Multiply $ \dfrac{4}{5} $ by $ b^2 $ to get $ \dfrac{ 4b^2 }{ 5 } $.

Step 1: Write $ b^2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{4}{5} \cdot b^2 & \xlongequal{\text{Step 1}} \frac{4}{5} \cdot \frac{b^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 4 \cdot b^2 }{ 5 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4b^2 }{ 5 } \end{aligned} $$

Multiply $ \dfrac{a}{2} $ by $ b $ to get $ \dfrac{ ab }{ 2 } $.

Step 1: Write $ b $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{a}{2} \cdot b & \xlongequal{\text{Step 1}} \frac{a}{2} \cdot \frac{b}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ a \cdot b }{ 2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ ab }{ 2 } \end{aligned} $$

Multiply $ \dfrac{2}{3} $ by $ a^2 $ to get $ \dfrac{ 2a^2 }{ 3 } $.

Step 1: Write $ a^2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{2}{3} \cdot a^2 & \xlongequal{\text{Step 1}} \frac{2}{3} \cdot \frac{a^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 2 \cdot a^2 }{ 3 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2a^2 }{ 3 } \end{aligned} $$

Multiply $ \dfrac{3a}{4} $ by $ b $ to get $ \dfrac{ 3ab }{ 4 } $.

Step 1: Write $ b $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{3a}{4} \cdot b & \xlongequal{\text{Step 1}} \frac{3a}{4} \cdot \frac{b}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 3a \cdot b }{ 4 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 3ab }{ 4 } \end{aligned} $$

Multiply $ \dfrac{4}{5} $ by $ b^2 $ to get $ \dfrac{ 4b^2 }{ 5 } $.

Step 1: Write $ b^2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{4}{5} \cdot b^2 & \xlongequal{\text{Step 1}} \frac{4}{5} \cdot \frac{b^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 4 \cdot b^2 }{ 5 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4b^2 }{ 5 } \end{aligned} $$

Multiply $ \dfrac{a}{2} $ by $ b $ to get $ \dfrac{ ab }{ 2 } $.

Step 1: Write $ b $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{a}{2} \cdot b & \xlongequal{\text{Step 1}} \frac{a}{2} \cdot \frac{b}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ a \cdot b }{ 2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ ab }{ 2 } \end{aligned} $$

Add $ \dfrac{2a^2}{3} $ and $ \dfrac{3ab}{4} $ to get $ \dfrac{ \color{purple}{ 8a^2+9ab } }{ 12 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ 4 }$ and the second by $\color{blue}{ 3 }$.

$$ \begin{aligned} \frac{2a^2}{3} + \frac{3ab}{4} & = \frac{ 2a^2 \cdot \color{blue}{ 4 }}{ 3 \cdot \color{blue}{ 4 }} + \frac{ 3ab \cdot \color{blue}{ 3 }}{ 4 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ 8a^2 } }{ 12 } + \frac{ \color{purple}{ 9ab } }{ 12 }=\frac{ \color{purple}{ 8a^2+9ab } }{ 12 } \end{aligned} $$

Multiply $ \dfrac{4}{5} $ by $ b^2 $ to get $ \dfrac{ 4b^2 }{ 5 } $.

Step 1: Write $ b^2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{4}{5} \cdot b^2 & \xlongequal{\text{Step 1}} \frac{4}{5} \cdot \frac{b^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 4 \cdot b^2 }{ 5 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4b^2 }{ 5 } \end{aligned} $$

Multiply $ \dfrac{a}{2} $ by $ b $ to get $ \dfrac{ ab }{ 2 } $.

Step 1: Write $ b $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{a}{2} \cdot b & \xlongequal{\text{Step 1}} \frac{a}{2} \cdot \frac{b}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ a \cdot b }{ 2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ ab }{ 2 } \end{aligned} $$

Add $ \dfrac{8a^2+9ab}{12} $ and $ \dfrac{4b^2}{5} $ to get $ \dfrac{ \color{purple}{ 40a^2+45ab+48b^2 } }{ 60 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ 5 }$ and the second by $\color{blue}{ 12 }$.

$$ \begin{aligned} \frac{8a^2+9ab}{12} + \frac{4b^2}{5} & = \frac{ \left( 8a^2+9ab \right) \cdot \color{blue}{ 5 }}{ 12 \cdot \color{blue}{ 5 }} + \frac{ 4b^2 \cdot \color{blue}{ 12 }}{ 5 \cdot \color{blue}{ 12 }} = \\[1ex] &=\frac{ \color{purple}{ 40a^2+45ab } }{ 60 } + \frac{ \color{purple}{ 48b^2 } }{ 60 }=\frac{ \color{purple}{ 40a^2+45ab+48b^2 } }{ 60 } \end{aligned} $$

Multiply $ \dfrac{ab}{2} $ by $ \dfrac{40a^2+45ab+48b^2}{60} $ to get $ \dfrac{ 40a^3b+45a^2b^2+48ab^3 }{ 120 } $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ab}{2} \cdot \frac{40a^2+45ab+48b^2}{60} & \xlongequal{\text{Step 1}} \frac{ ab \cdot \left( 40a^2+45ab+48b^2 \right) }{ 2 \cdot 60 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 40a^3b+45a^2b^2+48ab^3 }{ 120 } \end{aligned} $$