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Question

Answer

$$n^2+7n+\dfrac{6}{n^2}-n-42=\dfrac{n^4+6n^3-42n^2+6}{n^2}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}n^2+7n+\frac{6}{n^2}-n-42& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{n^4+7n^3+6}{n^2}-n-42 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{n^4+6n^3+6}{n^2}-42 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{n^4+6n^3-42n^2+6}{n^2}\end{aligned} $$
①

Add $n^2+7n$ and $ \dfrac{6}{n^2} $ to get $ \dfrac{ \color{purple}{ n^4+7n^3+6 } }{ n^2 }$.

Step 1: Write $ n^2+7n $ 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 $\color{blue}{ n^2 }$.

$$ \begin{aligned} n^2+7n+ \frac{6}{n^2} & \xlongequal{\text{Step 1}} \frac{n^2+7n}{\color{red}{1}} + \frac{6}{n^2} = \frac{ \left( n^2+7n \right) \cdot \color{blue}{ n^2 }}{ 1 \cdot \color{blue}{ n^2 }} + \frac{ 6 }{ n^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ n^4+7n^3 } }{ n^2 } + \frac{ \color{purple}{ 6 } }{ n^2 }=\frac{ \color{purple}{ n^4+7n^3+6 } }{ n^2 } \end{aligned} $$
②

Subtract $n$ from $ \dfrac{n^4+7n^3+6}{n^2} $ to get $ \dfrac{ \color{purple}{ n^4+6n^3+6 } }{ n^2 }$.

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

$$ \begin{aligned} \frac{n^4+7n^3+6}{n^2} -n & \xlongequal{\text{Step 1}} \frac{n^4+7n^3+6}{n^2} - \frac{n}{\color{red}{1}} = \frac{ n^4+7n^3+6 }{ n^2 } - \frac{ n \cdot \color{blue}{ n^2 }}{ 1 \cdot \color{blue}{ n^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ n^4+7n^3+6 } }{ n^2 } - \frac{ \color{purple}{ n^3 } }{ n^2 }=\frac{ \color{purple}{ n^4+6n^3+6 } }{ n^2 } \end{aligned} $$
③

Subtract $42$ from $ \dfrac{n^4+6n^3+6}{n^2} $ to get $ \dfrac{ \color{purple}{ n^4+6n^3-42n^2+6 } }{ n^2 }$.

Step 1: Write $ 42 $ 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 $\color{blue}{n^2}$.

$$ \begin{aligned} \frac{n^4+6n^3+6}{n^2} -42 & \xlongequal{\text{Step 1}} \frac{n^4+6n^3+6}{n^2} - \frac{42}{\color{red}{1}} = \frac{ n^4+6n^3+6 }{ n^2 } - \frac{ 42 \cdot \color{blue}{ n^2 }}{ 1 \cdot \color{blue}{ n^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ n^4+6n^3+6 } }{ n^2 } - \frac{ \color{purple}{ 42n^2 } }{ n^2 }=\frac{ \color{purple}{ n^4+6n^3-42n^2+6 } }{ n^2 } \end{aligned} $$
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