$$\dfrac{\dfrac{\dfrac{x}{2}x-4}{x-1}}{2x^2-2}=\dfrac{x^2-8}{4x^3-4x^2-4x+4}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\frac{\frac{x}{2}x-4}{x-1}}{2x^2-2}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\frac{\frac{x^2}{2}-4}{x-1}}{2x^2-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\frac{\frac{x^2-8}{2}}{x-1}}{2x^2-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{\frac{x^2-8}{2x-2}}{2x^2-2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{x^2-8}{4x^3-4x^2-4x+4}\end{aligned} $$ | |
| ① | Multiply $ \dfrac{x}{2} $ by $ x $ to get $ \dfrac{ x^2 }{ 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. $$ \begin{aligned} \frac{x}{2} \cdot x & \xlongequal{\text{Step 1}} \frac{x}{2} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ x \cdot x }{ 2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x^2 }{ 2 } \end{aligned} $$ |
| ② | Subtract $4$ from $ \dfrac{x^2}{2} $ to get $ \dfrac{ \color{purple}{ x^2-8 } }{ 2 }$. Step 1: Write $ 4 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: To subtract raitonal expressions, both fractions must have the same denominator. |
| ③ | Divide $ \dfrac{x^2-8}{2} $ by $ x-1 $ to get $ \dfrac{ x^2-8 }{ 2x-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. $$ \begin{aligned} \frac{ \frac{x^2-8}{2} }{x-1} & \xlongequal{\text{Step 1}} \frac{x^2-8}{2} \cdot \frac{\color{blue}{1}}{\color{blue}{x-1}} \xlongequal{\text{Step 2}} \frac{ \left( x^2-8 \right) \cdot 1 }{ 2 \cdot \left( x-1 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^2-8 }{ 2x-2 } \end{aligned} $$ |
| ④ | Divide $ \dfrac{x^2-8}{2x-2} $ by $ 2x^2-2 $ to get $ \dfrac{x^2-8}{4x^3-4x^2-4x+4} $. 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. $$ \begin{aligned} \frac{ \frac{x^2-8}{2x-2} }{2x^2-2} & \xlongequal{\text{Step 1}} \frac{x^2-8}{2x-2} \cdot \frac{\color{blue}{1}}{\color{blue}{2x^2-2}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^2-8 \right) \cdot 1 }{ \left( 2x-2 \right) \cdot \left( 2x^2-2 \right) } \xlongequal{\text{Step 3}} \frac{ x^2-8 }{ 4x^3-4x-4x^2+4 } = \\[1ex] &= \frac{x^2-8}{4x^3-4x^2-4x+4} \end{aligned} $$ |