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Question

Answer

$$n+\dfrac{8}{n^2}+n-56=\dfrac{2n^3-56n^2+8}{n^2}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}n+\frac{8}{n^2}+n-56& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{n^3+8}{n^2}+n-56 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2n^3+8}{n^2}-56 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2n^3-56n^2+8}{n^2}\end{aligned} $$
①

Add $n$ and $ \dfrac{8}{n^2} $ to get $ \dfrac{ \color{purple}{ n^3+8 } }{ n^2 }$.

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

$$ \begin{aligned} n+ \frac{8}{n^2} & \xlongequal{\text{Step 1}} \frac{n}{\color{red}{1}} + \frac{8}{n^2} = \frac{ n \cdot \color{blue}{ n^2 }}{ 1 \cdot \color{blue}{ n^2 }} + \frac{ 8 }{ n^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ n^3 } }{ n^2 } + \frac{ \color{purple}{ 8 } }{ n^2 }=\frac{ \color{purple}{ n^3+8 } }{ n^2 } \end{aligned} $$
②

Add $ \dfrac{n^3+8}{n^2} $ and $ n $ to get $ \dfrac{ \color{purple}{ 2n^3+8 } }{ n^2 }$.

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

$$ \begin{aligned} \frac{n^3+8}{n^2} +n & \xlongequal{\text{Step 1}} \frac{n^3+8}{n^2} + \frac{n}{\color{red}{1}} = \frac{ n^3+8 }{ n^2 } + \frac{ n \cdot \color{blue}{ n^2 }}{ 1 \cdot \color{blue}{ n^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ n^3+8 } }{ n^2 } + \frac{ \color{purple}{ n^3 } }{ n^2 }=\frac{ \color{purple}{ 2n^3+8 } }{ n^2 } \end{aligned} $$
③

Subtract $56$ from $ \dfrac{2n^3+8}{n^2} $ to get $ \dfrac{ \color{purple}{ 2n^3-56n^2+8 } }{ n^2 }$.

Step 1: Write $ 56 $ 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 $\color{blue}{n^2}$.

$$ \begin{aligned} \frac{2n^3+8}{n^2} -56 & \xlongequal{\text{Step 1}} \frac{2n^3+8}{n^2} - \frac{56}{\color{red}{1}} = \frac{ 2n^3+8 }{ n^2 } - \frac{ 56 \cdot \color{blue}{ n^2 }}{ 1 \cdot \color{blue}{ n^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2n^3+8 } }{ n^2 } - \frac{ \color{purple}{ 56n^2 } }{ n^2 }=\frac{ \color{purple}{ 2n^3-56n^2+8 } }{ n^2 } \end{aligned} $$
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