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Question

Answer

$$a^2b-a\dfrac{b^2}{a^2}-ab=\dfrac{a^4b-a^3b-ab^2}{a^2}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}a^2b-a\frac{b^2}{a^2}-ab& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}a^2b-\frac{ab^2}{a^2}-ab \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{a^4b-ab^2}{a^2}-ab \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{a^4b-a^3b-ab^2}{a^2}\end{aligned} $$
①

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

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

Subtract $ \dfrac{ab^2}{a^2} $ from $ a^2b $ to get $ \dfrac{ \color{purple}{ a^4b-ab^2 } }{ a^2 }$.

Step 1: Write $ a^2b $ 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 $\color{blue}{ a^2 }$.

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

Subtract $ab$ from $ \dfrac{a^4b-ab^2}{a^2} $ to get $ \dfrac{ \color{purple}{ a^4b-a^3b-ab^2 } }{ a^2 }$.

Step 1: Write $ ab $ 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 second fraction by $\color{blue}{a^2}$.

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