Multiply $100$ by $ \dfrac{a}{a} $ to get $ \dfrac{ 100a }{ a } $.

Step 1: Write $ 100 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 100 \cdot \frac{a}{a} & \xlongequal{\text{Step 1}} \frac{100}{\color{red}{1}} \cdot \frac{a}{a} \xlongequal{\text{Step 2}} \frac{ 100 \cdot a }{ 1 \cdot a } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 100a }{ a } \end{aligned} $$

Add $10a^2$ and $ \dfrac{100a}{a} $ to get $ \dfrac{ \color{purple}{ 10a^3+100a } }{ a }$.

Step 1: Write $ 10a^2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ a }$.

$$ \begin{aligned} 10a^2+ \frac{100a}{a} & \xlongequal{\text{Step 1}} \frac{10a^2}{\color{red}{1}} + \frac{100a}{a} = \frac{ 10a^2 \cdot \color{blue}{ a }}{ 1 \cdot \color{blue}{ a }} + \frac{ 100a }{ a } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 10a^3 } }{ a } + \frac{ \color{purple}{ 100a } }{ a }=\frac{ \color{purple}{ 10a^3+100a } }{ a } \end{aligned} $$

Add $ \dfrac{10a^3+100a}{a} $ and $ 10 $ to get $ \dfrac{ \color{purple}{ 10a^3+110a } }{ a }$.

Step 1: Write $ 10 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{a}$.

$$ \begin{aligned} \frac{10a^3+100a}{a} +10 & \xlongequal{\text{Step 1}} \frac{10a^3+100a}{a} + \frac{10}{\color{red}{1}} = \frac{ 10a^3+100a }{ a } + \frac{ 10 \cdot \color{blue}{ a }}{ 1 \cdot \color{blue}{ a }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 10a^3+100a } }{ a } + \frac{ \color{purple}{ 10a } }{ a }=\frac{ \color{purple}{ 10a^3+110a } }{ a } \end{aligned} $$