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Question

Answer

$$(i\dfrac{w}{3}+1)(-\dfrac{w^2}{24}+i\dfrac{w}{8}+1)=\dfrac{3i^2w^2-iw^3+33iw-3w^2+72}{72}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}(i\frac{w}{3}+1)(-\frac{w^2}{24}+i\frac{w}{8}+1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(\frac{iw}{3}+1)(-\frac{w^2}{24}+\frac{iw}{8}+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{iw+3}{3}(\frac{3iw-w^2}{24}+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{iw+3}{3}\frac{3iw-w^2+24}{24} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{3i^2w^2-iw^3+33iw-3w^2+72}{72}\end{aligned} $$
①

Multiply $i$ by $ \dfrac{w}{3} $ to get $ \dfrac{ iw }{ 3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} i \cdot \frac{w}{3} & \xlongequal{\text{Step 1}} \frac{i}{\color{red}{1}} \cdot \frac{w}{3} \xlongequal{\text{Step 2}} \frac{ i \cdot w }{ 1 \cdot 3 } \xlongequal{\text{Step 3}} \frac{ iw }{ 3 } \end{aligned} $$
②

Multiply $i$ by $ \dfrac{w}{8} $ to get $ \dfrac{ iw }{ 8 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} i \cdot \frac{w}{8} & \xlongequal{\text{Step 1}} \frac{i}{\color{red}{1}} \cdot \frac{w}{8} \xlongequal{\text{Step 2}} \frac{ i \cdot w }{ 1 \cdot 8 } \xlongequal{\text{Step 3}} \frac{ iw }{ 8 } \end{aligned} $$
③

Add $ \dfrac{iw}{3} $ and $ 1 $ to get $ \dfrac{ \color{purple}{ iw+3 } }{ 3 }$.

Step 1: Write $ 1 $ 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 $\color{blue}{3}$.

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

Add $ \dfrac{-w^2}{24} $ and $ \dfrac{iw}{8} $ to get $ \dfrac{ \color{purple}{ 3iw-w^2 } }{ 24 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{3}$.

$$ \begin{aligned} \frac{-w^2}{24} + \frac{iw}{8} & = \frac{ -w^2 }{ 24 } + \frac{ iw \cdot \color{blue}{ 3 }}{ 8 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ -w^2 } }{ 24 } + \frac{ \color{purple}{ 3iw } }{ 24 }=\frac{ \color{purple}{ 3iw-w^2 } }{ 24 } \end{aligned} $$
⑤

Add $ \dfrac{iw}{3} $ and $ 1 $ to get $ \dfrac{ \color{purple}{ iw+3 } }{ 3 }$.

Step 1: Write $ 1 $ 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 $\color{blue}{3}$.

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

Add $ \dfrac{3iw-w^2}{24} $ and $ 1 $ to get $ \dfrac{ \color{purple}{ 3iw-w^2+24 } }{ 24 }$.

Step 1: Write $ 1 $ 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 $\color{blue}{24}$.

$$ \begin{aligned} \frac{3iw-w^2}{24} +1 & \xlongequal{\text{Step 1}} \frac{3iw-w^2}{24} + \frac{1}{\color{red}{1}} = \frac{ 3iw-w^2 }{ 24 } + \frac{ 1 \cdot \color{blue}{ 24 }}{ 1 \cdot \color{blue}{ 24 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 3iw-w^2 } }{ 24 } + \frac{ \color{purple}{ 24 } }{ 24 }=\frac{ \color{purple}{ 3iw-w^2+24 } }{ 24 } \end{aligned} $$
⑦

Multiply $ \dfrac{iw+3}{3} $ by $ \dfrac{3iw-w^2+24}{24} $ to get $ \dfrac{3i^2w^2-iw^3+33iw-3w^2+72}{72} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{iw+3}{3} \cdot \frac{3iw-w^2+24}{24} & \xlongequal{\text{Step 1}} \frac{ \left( iw+3 \right) \cdot \left( 3iw-w^2+24 \right) }{ 3 \cdot 24 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 3i^2w^2-iw^3+24iw+9iw-3w^2+72 }{ 72 } = \frac{3i^2w^2-iw^3+33iw-3w^2+72}{72} \end{aligned} $$
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