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Question

Answer

$$4w^2+24w+\dfrac{32}{w^2}-4w-32=\dfrac{4w^4+20w^3-32w^2+32}{w^2}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}4w^2+24w+\frac{32}{w^2}-4w-32& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4w^4+24w^3+32}{w^2}-4w-32 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4w^4+20w^3+32}{w^2}-32 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{4w^4+20w^3-32w^2+32}{w^2}\end{aligned} $$
①

Add $4w^2+24w$ and $ \dfrac{32}{w^2} $ to get $ \dfrac{ \color{purple}{ 4w^4+24w^3+32 } }{ w^2 }$.

Step 1: Write $ 4w^2+24w $ 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 $\color{blue}{ w^2 }$.

$$ \begin{aligned} 4w^2+24w+ \frac{32}{w^2} & \xlongequal{\text{Step 1}} \frac{4w^2+24w}{\color{red}{1}} + \frac{32}{w^2} = \frac{ \left( 4w^2+24w \right) \cdot \color{blue}{ w^2 }}{ 1 \cdot \color{blue}{ w^2 }} + \frac{ 32 }{ w^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4w^4+24w^3 } }{ w^2 } + \frac{ \color{purple}{ 32 } }{ w^2 }=\frac{ \color{purple}{ 4w^4+24w^3+32 } }{ w^2 } \end{aligned} $$
②

Subtract $4w$ from $ \dfrac{4w^4+24w^3+32}{w^2} $ to get $ \dfrac{ \color{purple}{ 4w^4+20w^3+32 } }{ w^2 }$.

Step 1: Write $ 4w $ 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 $\color{blue}{w^2}$.

$$ \begin{aligned} \frac{4w^4+24w^3+32}{w^2} -4w & \xlongequal{\text{Step 1}} \frac{4w^4+24w^3+32}{w^2} - \frac{4w}{\color{red}{1}} = \frac{ 4w^4+24w^3+32 }{ w^2 } - \frac{ 4w \cdot \color{blue}{ w^2 }}{ 1 \cdot \color{blue}{ w^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4w^4+24w^3+32 } }{ w^2 } - \frac{ \color{purple}{ 4w^3 } }{ w^2 }=\frac{ \color{purple}{ 4w^4+20w^3+32 } }{ w^2 } \end{aligned} $$
③

Subtract $32$ from $ \dfrac{4w^4+20w^3+32}{w^2} $ to get $ \dfrac{ \color{purple}{ 4w^4+20w^3-32w^2+32 } }{ w^2 }$.

Step 1: Write $ 32 $ 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 $\color{blue}{w^2}$.

$$ \begin{aligned} \frac{4w^4+20w^3+32}{w^2} -32 & \xlongequal{\text{Step 1}} \frac{4w^4+20w^3+32}{w^2} - \frac{32}{\color{red}{1}} = \frac{ 4w^4+20w^3+32 }{ w^2 } - \frac{ 32 \cdot \color{blue}{ w^2 }}{ 1 \cdot \color{blue}{ w^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4w^4+20w^3+32 } }{ w^2 } - \frac{ \color{purple}{ 32w^2 } }{ w^2 }=\frac{ \color{purple}{ 4w^4+20w^3-32w^2+32 } }{ w^2 } \end{aligned} $$
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