$$\dfrac{1}{2}x+\dfrac{1}{4}\dfrac{\dfrac{x}{1}}{2}x^2=\dfrac{x^3+4x}{8}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{2}x+\frac{1}{4}\frac{\frac{x}{1}}{2}x^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{x}{2}+\frac{1}{4}\frac{x}{2}x^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{x}{2}+\frac{x}{8}x^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{x}{2}+\frac{x^3}{8} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{x^3+4x}{8}\end{aligned} $$ | |
| ① | Multiply $ \dfrac{1}{2} $ by $ x $ to get $ \dfrac{ x }{ 2 } $. Step 1: Write $ x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{1}{2} \cdot x & \xlongequal{\text{Step 1}} \frac{1}{2} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot x }{ 2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x }{ 2 } \end{aligned} $$ |
| ② | Divide $ \dfrac{x}{1} $ by $ 2 $ to get $ \dfrac{ x }{ 2 } $. Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{ \frac{x}{1} }{2} & \xlongequal{\text{Step 1}} \frac{x}{1} \cdot \frac{\color{blue}{1}}{\color{blue}{2}} \xlongequal{\text{Step 2}} \frac{ x \cdot 1 }{ 1 \cdot 2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x }{ 2 } \end{aligned} $$ |
| ③ | Multiply $ \dfrac{1}{2} $ by $ x $ to get $ \dfrac{ x }{ 2 } $. Step 1: Write $ x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{1}{2} \cdot x & \xlongequal{\text{Step 1}} \frac{1}{2} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot x }{ 2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x }{ 2 } \end{aligned} $$ |
| ④ | Multiply $ \dfrac{1}{4} $ by $ \dfrac{x}{2} $ to get $ \dfrac{ x }{ 8 } $. Step 1: Multiply numerators and denominators. Step 2: Simplify numerator and denominator. $$ \begin{aligned} \frac{1}{4} \cdot \frac{x}{2} \xlongequal{\text{Step 1}} \frac{ 1 \cdot x }{ 4 \cdot 2 } \xlongequal{\text{Step 2}} \frac{ x }{ 8 } \end{aligned} $$ |
| ⑤ | Multiply $ \dfrac{1}{2} $ by $ x $ to get $ \dfrac{ x }{ 2 } $. Step 1: Write $ x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{1}{2} \cdot x & \xlongequal{\text{Step 1}} \frac{1}{2} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot x }{ 2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x }{ 2 } \end{aligned} $$ |
| ⑥ | Multiply $ \dfrac{x}{8} $ by $ x^2 $ to get $ \dfrac{ x^3 }{ 8 } $. Step 1: Write $ x^2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{x}{8} \cdot x^2 & \xlongequal{\text{Step 1}} \frac{x}{8} \cdot \frac{x^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ x \cdot x^2 }{ 8 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^3 }{ 8 } \end{aligned} $$ |
| ⑦ | Add $ \dfrac{x}{2} $ and $ \dfrac{x^3}{8} $ to get $ \dfrac{ \color{purple}{ x^3+4x } }{ 8 }$. To add raitonal expressions, both fractions must have the same denominator. |