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Question

Answer

$$\dfrac{3}{a}+\dfrac{1}{6}\dfrac{a}{2}a+2=\dfrac{a^3+24a+36}{12a}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}\frac{3}{a}+\frac{1}{6}\frac{a}{2}a+2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3}{a}+\frac{a}{12}a+2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{3}{a}+\frac{a^2}{12}+2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{a^3+36}{12a}+2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{a^3+24a+36}{12a}\end{aligned} $$
①

Multiply $ \dfrac{1}{6} $ by $ \dfrac{a}{2} $ to get $ \dfrac{ a }{ 12 } $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{6} \cdot \frac{a}{2} \xlongequal{\text{Step 1}} \frac{ 1 \cdot a }{ 6 \cdot 2 } \xlongequal{\text{Step 2}} \frac{ a }{ 12 } \end{aligned} $$
②

Multiply $ \dfrac{a}{12} $ by $ a $ to get $ \dfrac{ a^2 }{ 12 } $.

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.

$$ \begin{aligned} \frac{a}{12} \cdot a & \xlongequal{\text{Step 1}} \frac{a}{12} \cdot \frac{a}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ a \cdot a }{ 12 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ a^2 }{ 12 } \end{aligned} $$
③

Add $ \dfrac{3}{a} $ and $ \dfrac{a^2}{12} $ to get $ \dfrac{ \color{purple}{ a^3+36 } }{ 12a }$.

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}{ 12 }$ and the second by $\color{blue}{ a }$.

$$ \begin{aligned} \frac{3}{a} + \frac{a^2}{12} & = \frac{ 3 \cdot \color{blue}{ 12 }}{ a \cdot \color{blue}{ 12 }} + \frac{ a^2 \cdot \color{blue}{ a }}{ 12 \cdot \color{blue}{ a }} = \\[1ex] &=\frac{ \color{purple}{ 36 } }{ 12a } + \frac{ \color{purple}{ a^3 } }{ 12a }=\frac{ \color{purple}{ a^3+36 } }{ 12a } \end{aligned} $$
④

Add $ \dfrac{a^3+36}{12a} $ and $ 2 $ to get $ \dfrac{ \color{purple}{ a^3+24a+36 } }{ 12a }$.

Step 1: Write $ 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 second fraction by $\color{blue}{12a}$.

$$ \begin{aligned} \frac{a^3+36}{12a} +2 & \xlongequal{\text{Step 1}} \frac{a^3+36}{12a} + \frac{2}{\color{red}{1}} = \frac{ a^3+36 }{ 12a } + \frac{ 2 \cdot \color{blue}{ 12a }}{ 1 \cdot \color{blue}{ 12a }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ a^3+36 } }{ 12a } + \frac{ \color{purple}{ 24a } }{ 12a }=\frac{ \color{purple}{ a^3+24a+36 } }{ 12a } \end{aligned} $$
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