$$(j\dfrac{w}{3}+1)(-\dfrac{w^2}{24}+j\dfrac{w}{8}+1)=\dfrac{3j^2w^2-jw^3+33jw-3w^2+72}{72}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(j\frac{w}{3}+1)(-\frac{w^2}{24}+j\frac{w}{8}+1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(\frac{jw}{3}+1)(-\frac{w^2}{24}+\frac{jw}{8}+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{jw+3}{3}(\frac{3jw-w^2}{24}+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{jw+3}{3}\frac{3jw-w^2+24}{24} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{3j^2w^2-jw^3+33jw-3w^2+72}{72}\end{aligned} $$ | |
| ① | Multiply $j$ by $ \dfrac{w}{3} $ to get $ \dfrac{ jw }{ 3 } $. Step 1: Write $ j $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} j \cdot \frac{w}{3} & \xlongequal{\text{Step 1}} \frac{j}{\color{red}{1}} \cdot \frac{w}{3} \xlongequal{\text{Step 2}} \frac{ j \cdot w }{ 1 \cdot 3 } \xlongequal{\text{Step 3}} \frac{ jw }{ 3 } \end{aligned} $$ |
| ② | Multiply $j$ by $ \dfrac{w}{8} $ to get $ \dfrac{ jw }{ 8 } $. Step 1: Write $ j $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} j \cdot \frac{w}{8} & \xlongequal{\text{Step 1}} \frac{j}{\color{red}{1}} \cdot \frac{w}{8} \xlongequal{\text{Step 2}} \frac{ j \cdot w }{ 1 \cdot 8 } \xlongequal{\text{Step 3}} \frac{ jw }{ 8 } \end{aligned} $$ |
| ③ | Add $ \dfrac{jw}{3} $ and $ 1 $ to get $ \dfrac{ \color{purple}{ jw+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. |
| ④ | Add $ \dfrac{-w^2}{24} $ and $ \dfrac{jw}{8} $ to get $ \dfrac{ \color{purple}{ 3jw-w^2 } }{ 24 }$. To add raitonal expressions, both fractions must have the same denominator. |
| ⑤ | Add $ \dfrac{jw}{3} $ and $ 1 $ to get $ \dfrac{ \color{purple}{ jw+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. |
| ⑥ | Add $ \dfrac{3jw-w^2}{24} $ and $ 1 $ to get $ \dfrac{ \color{purple}{ 3jw-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. |
| ⑦ | Multiply $ \dfrac{jw+3}{3} $ by $ \dfrac{3jw-w^2+24}{24} $ to get $ \dfrac{3j^2w^2-jw^3+33jw-3w^2+72}{72} $. Step 1: Multiply numerators and denominators. Step 2: Simplify numerator and denominator. $$ \begin{aligned} \frac{jw+3}{3} \cdot \frac{3jw-w^2+24}{24} & \xlongequal{\text{Step 1}} \frac{ \left( jw+3 \right) \cdot \left( 3jw-w^2+24 \right) }{ 3 \cdot 24 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 3j^2w^2-jw^3+24jw+9jw-3w^2+72 }{ 72 } = \frac{3j^2w^2-jw^3+33jw-3w^2+72}{72} \end{aligned} $$ |