$$\dfrac{\dfrac{x^2-16}{x^3-6x^2+8x}\dfrac{2-x}{x^2-100}}{\dfrac{x^2+12x+32}{x^2+18x+80}}=-\dfrac{1}{x^2-10x}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\frac{x^2-16}{x^3-6x^2+8x}\frac{2-x}{x^2-100}}{\frac{x^2+12x+32}{x^2+18x+80}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\frac{x+4}{x^2-2x}\frac{2-x}{x^2-100}}{\frac{x^2+12x+32}{x^2+18x+80}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\frac{-x-4}{x^3-100x}}{\frac{x^2+12x+32}{x^2+18x+80}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{\frac{-x-4}{x^3-100x}}{\frac{x+4}{x+10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{1}{x^2-10x}\end{aligned} $$ | |
| ① | Simplify $ \dfrac{x^2-16}{x^3-6x^2+8x} $ to $ \dfrac{x+4}{x^2-2x} $. Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{x-4}$. $$ \begin{aligned} \frac{x^2-16}{x^3-6x^2+8x} & =\frac{ \left( x+4 \right) \cdot \color{blue}{ \left( x-4 \right) }}{ \left( x^2-2x \right) \cdot \color{blue}{ \left( x-4 \right) }} = \\[1ex] &= \frac{x+4}{x^2-2x} \end{aligned} $$ |
| ② | Multiply $ \dfrac{x+4}{x^2-2x} $ by $ \dfrac{2-x}{x^2-100} $ to get $ \dfrac{ -x-4 }{ x^3-100x } $. Step 1: Factor numerators and denominators. Step 2: Cancel common factors. Step 3: Multiply numerators and denominators. Step 4: Simplify numerator and denominator. $$ \begin{aligned} \frac{x+4}{x^2-2x} \cdot \frac{2-x}{x^2-100} & \xlongequal{\text{Step 1}} \frac{ x+4 }{ x \cdot \color{red}{ \left( x-2 \right) } } \cdot \frac{ \left( -1 \right) \cdot \color{red}{ \left( x-2 \right) } }{ x^2-100 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ x+4 }{ x } \cdot \frac{ -1 }{ x^2-100 } \xlongequal{\text{Step 3}} \frac{ \left( x+4 \right) \cdot \left( -1 \right) }{ x \cdot \left( x^2-100 \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ -x-4 }{ x^3-100x } \end{aligned} $$ |
| ③ | Simplify $ \dfrac{x^2+12x+32}{x^2+18x+80} $ to $ \dfrac{x+4}{x+10} $. Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{x+8}$. $$ \begin{aligned} \frac{x^2+12x+32}{x^2+18x+80} & =\frac{ \left( x+4 \right) \cdot \color{blue}{ \left( x+8 \right) }}{ \left( x+10 \right) \cdot \color{blue}{ \left( x+8 \right) }} = \\[1ex] &= \frac{x+4}{x+10} \end{aligned} $$ |
| ④ | Divide $ \dfrac{-x-4}{x^3-100x} $ by $ \dfrac{x+4}{x+10} $ to get $ \dfrac{ -1 }{ x^2-10x } $. Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction. Step 2: Factor numerators and denominators. Step 3: Cancel common factors. Step 4: Multiply numerators and denominators. Step 5: Simplify numerator and denominator. $$ \begin{aligned} \frac{ \frac{-x-4}{x^3-100x} }{ \frac{\color{blue}{x+4}}{\color{blue}{x+10}} } & \xlongequal{\text{Step 1}} \frac{-x-4}{x^3-100x} \cdot \frac{\color{blue}{x+10}}{\color{blue}{x+4}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( -1 \right) \cdot \color{blue}{ \left( x+4 \right) } }{ \left( x^2-10x \right) \cdot \color{red}{ \left( x+10 \right) } } \cdot \frac{ 1 \cdot \color{red}{ \left( x+10 \right) } }{ 1 \cdot \color{blue}{ \left( x+4 \right) } } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ -1 }{ x^2-10x } \cdot \frac{ 1 }{ 1 } \xlongequal{\text{Step 4}} \frac{ \left( -1 \right) \cdot 1 }{ \left( x^2-10x \right) \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ -1 }{ x^2-10x } \end{aligned} $$ |