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Question

Answer

$$6r^2+42\cdot\dfrac{r}{r^2}+r-42=\dfrac{6r^4+r^3-42r^2+42r}{r^2}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}6r^2+42 \cdot \frac{r}{r^2}+r-42& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}6r^2+\frac{42r}{r^2}+r-42 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{6r^4+42r}{r^2}+r-42 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{6r^4+r^3+42r}{r^2}-42 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{6r^4+r^3-42r^2+42r}{r^2}\end{aligned} $$
①

Multiply $42$ by $ \dfrac{r}{r^2} $ to get $ \dfrac{ 42r }{ r^2 } $.

Step 1: Write $ 42 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 42 \cdot \frac{r}{r^2} & \xlongequal{\text{Step 1}} \frac{42}{\color{red}{1}} \cdot \frac{r}{r^2} \xlongequal{\text{Step 2}} \frac{ 42 \cdot r }{ 1 \cdot r^2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 42r }{ r^2 } \end{aligned} $$
②

Add $6r^2$ and $ \dfrac{42r}{r^2} $ to get $ \dfrac{ \color{purple}{ 6r^4+42r } }{ r^2 }$.

Step 1: Write $ 6r^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 first fraction by $\color{blue}{ r^2 }$.

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

Add $ \dfrac{6r^4+42r}{r^2} $ and $ r $ to get $ \dfrac{ \color{purple}{ 6r^4+r^3+42r } }{ r^2 }$.

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

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

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

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