Use the distributive property.

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

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

Step 2: Cancel $ \color{blue}{ a } $ in first and second fraction.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

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

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

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

Step 2: Cancel $ \color{blue}{ a } $ in first and second fraction.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

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

Multiply $ab$ by $ \dfrac{c}{a} $ to get $ \dfrac{ abc }{ a } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

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

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

Step 2: Cancel $ \color{blue}{ a } $ in first and second fraction.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

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

Subtract $ \dfrac{abc}{a} $ from $ b^2 $ to get $ \dfrac{ \color{purple}{ ab^2-abc } }{ a }$.

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

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

$$ \begin{aligned} b^2- \frac{abc}{a} & \xlongequal{\text{Step 1}} \frac{b^2}{\color{red}{1}} - \frac{abc}{a} = \frac{ b^2 \cdot \color{blue}{ a }}{ 1 \cdot \color{blue}{ a }} - \frac{ abc }{ a } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ ab^2 } }{ a } - \frac{ \color{purple}{ abc } }{ a }=\frac{ \color{purple}{ ab^2-abc } }{ a } \end{aligned} $$

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

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

Step 2: Cancel $ \color{blue}{ a } $ in first and second fraction.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

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

Multiply $ \dfrac{ab^2-abc}{a} $ by $ b $ to get $ \dfrac{ ab^3-ab^2c }{ a } $.

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.

$$ \begin{aligned} \frac{ab^2-abc}{a} \cdot b & \xlongequal{\text{Step 1}} \frac{ab^2-abc}{a} \cdot \frac{b}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( ab^2-abc \right) \cdot b }{ a \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ ab^3-ab^2c }{ a } \end{aligned} $$

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

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

Step 2: Cancel $ \color{blue}{ a } $ in first and second fraction.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

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

$$ 1 b b = b^{1 + 1} = b^2 $$

Add $ \dfrac{ab^3-ab^2c}{a} $ and $ b^2 $ to get $ \dfrac{ \color{purple}{ ab^3-ab^2c+ab^2 } }{ a }$.

Step 1: Write $ b^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 second fraction by $\color{blue}{a}$.

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