$$(x^2-9)\dfrac{x^2+3a+9}{x^4}x^{10}(x^3-81)=\dfrac{x^{17}+3ax^{15}-27ax^{13}-81x^{14}-243ax^{12}-81x^{13}+2187ax^{10}+6561x^{10}}{x^4}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x^2-9)\frac{x^2+3a+9}{x^4}x^{10}(x^3-81)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{x^4+3ax^2-27a-81}{x^4}x^{10}(x^3-81) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{x^{14}+3ax^{12}-27ax^{10}-81x^{10}}{x^4}(x^3-81) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{x^{17}+3ax^{15}-27ax^{13}-81x^{14}-243ax^{12}-81x^{13}+2187ax^{10}+6561x^{10}}{x^4}\end{aligned} $$ | |
| ① | Multiply $x^2-9$ by $ \dfrac{x^2+3a+9}{x^4} $ to get $ \dfrac{x^4+3ax^2-27a-81}{x^4} $. Step 1: Write $ x^2-9 $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} x^2-9 \cdot \frac{x^2+3a+9}{x^4} & \xlongequal{\text{Step 1}} \frac{x^2-9}{\color{red}{1}} \cdot \frac{x^2+3a+9}{x^4} \xlongequal{\text{Step 2}} \frac{ \left( x^2-9 \right) \cdot \left( x^2+3a+9 \right) }{ 1 \cdot x^4 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^4+3ax^2+ \cancel{9x^2} -\cancel{9x^2}-27a-81 }{ x^4 } = \frac{x^4+3ax^2-27a-81}{x^4} \end{aligned} $$ |
| ② | Multiply $ \dfrac{x^4+3ax^2-27a-81}{x^4} $ by $ x^{10} $ to get $ \dfrac{ x^{14}+3ax^{12}-27ax^{10}-81x^{10} }{ x^4 } $. Step 1: Write $ x^{10} $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{x^4+3ax^2-27a-81}{x^4} \cdot x^{10} & \xlongequal{\text{Step 1}} \frac{x^4+3ax^2-27a-81}{x^4} \cdot \frac{x^{10}}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^4+3ax^2-27a-81 \right) \cdot x^{10} }{ x^4 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x^{14}+3ax^{12}-27ax^{10}-81x^{10} }{ x^4 } \end{aligned} $$ |
| ③ | Multiply $ \dfrac{x^{14}+3ax^{12}-27ax^{10}-81x^{10}}{x^4} $ by $ x^3-81 $ to get $ \dfrac{x^{17}+3ax^{15}-27ax^{13}-81x^{14}-243ax^{12}-81x^{13}+2187ax^{10}+6561x^{10}}{x^4} $. Step 1: Write $ x^3-81 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{x^{14}+3ax^{12}-27ax^{10}-81x^{10}}{x^4} \cdot x^3-81 & \xlongequal{\text{Step 1}} \frac{x^{14}+3ax^{12}-27ax^{10}-81x^{10}}{x^4} \cdot \frac{x^3-81}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^{14}+3ax^{12}-27ax^{10}-81x^{10} \right) \cdot \left( x^3-81 \right) }{ x^4 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^{17}-81x^{14}+3ax^{15}-243ax^{12}-27ax^{13}+2187ax^{10}-81x^{13}+6561x^{10} }{ x^4 } = \frac{x^{17}+3ax^{15}-27ax^{13}-81x^{14}-243ax^{12}-81x^{13}+2187ax^{10}+6561x^{10}}{x^4} \end{aligned} $$ |