$$24\cdot\dfrac{\dfrac{\dfrac{x^2}{2x-4}}{12x^2+36x}}{x^2-11x+18}=\dfrac{24x^2}{24x^5-240x^4+24x^3+2016x^2-2592x}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}24 \cdot \frac{\frac{\frac{x^2}{2x-4}}{12x^2+36x}}{x^2-11x+18}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}24 \cdot \frac{\frac{x^2}{24x^3+24x^2-144x}}{x^2-11x+18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}24 \cdot \frac{x^2}{24x^5-240x^4+24x^3+2016x^2-2592x} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{24x^2}{24x^5-240x^4+24x^3+2016x^2-2592x}\end{aligned} $$ | |
| ① | Divide $ \dfrac{x^2}{2x-4} $ by $ 12x^2+36x $ to get $ \dfrac{x^2}{24x^3+24x^2-144x} $. 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. $$ \begin{aligned} \frac{ \frac{x^2}{2x-4} }{12x^2+36x} & \xlongequal{\text{Step 1}} \frac{x^2}{2x-4} \cdot \frac{\color{blue}{1}}{\color{blue}{12x^2+36x}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ x^2 \cdot 1 }{ \left( 2x-4 \right) \cdot \left( 12x^2+36x \right) } \xlongequal{\text{Step 3}} \frac{ x^2 }{ 24x^3+72x^2-48x^2-144x } = \\[1ex] &= \frac{x^2}{24x^3+24x^2-144x} \end{aligned} $$ |
| ② | Divide $ \dfrac{x^2}{24x^3+24x^2-144x} $ by $ x^2-11x+18 $ to get $ \dfrac{x^2}{24x^5-240x^4+24x^3+2016x^2-2592x} $. 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. $$ \begin{aligned} \frac{ \frac{x^2}{24x^3+24x^2-144x} }{x^2-11x+18} & \xlongequal{\text{Step 1}} \frac{x^2}{24x^3+24x^2-144x} \cdot \frac{\color{blue}{1}}{\color{blue}{x^2-11x+18}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ x^2 \cdot 1 }{ \left( 24x^3+24x^2-144x \right) \cdot \left( x^2-11x+18 \right) } \xlongequal{\text{Step 3}} \frac{ x^2 }{ 24x^5-264x^4+432x^3+24x^4-264x^3+432x^2-144x^3+1584x^2-2592x } = \\[1ex] &= \frac{x^2}{24x^5-240x^4+24x^3+2016x^2-2592x} \end{aligned} $$ |
| ③ | Multiply $24$ by $ \dfrac{x^2}{24x^5-240x^4+24x^3+2016x^2-2592x} $ to get $ \dfrac{ 24x^2 }{ 24x^5-240x^4+24x^3+2016x^2-2592x } $. Step 1: Write $ 24 $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} 24 \cdot \frac{x^2}{24x^5-240x^4+24x^3+2016x^2-2592x} & \xlongequal{\text{Step 1}} \frac{24}{\color{red}{1}} \cdot \frac{x^2}{24x^5-240x^4+24x^3+2016x^2-2592x} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 24 \cdot x^2 }{ 1 \cdot \left( 24x^5-240x^4+24x^3+2016x^2-2592x \right) } \xlongequal{\text{Step 3}} \frac{ 24x^2 }{ 24x^5-240x^4+24x^3+2016x^2-2592x } \end{aligned} $$ |