Answer
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{9x^2+27x+18}{x^2+8x+12}\frac{\frac{x+6}{x+1}}{\frac{x^2+7x+6}{x^2+7x+6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{9x+9}{x+6}\frac{\frac{x+6}{x+1}}{\frac{x^2+7x+6}{x^2+7x+6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{9x+9}{x+6}\frac{\frac{x+6}{x+1}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{9x+9}{x+6}\frac{x+6}{x+1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}9\end{aligned} $$ | |
| ① | Simplify $ \dfrac{9x^2+27x+18}{x^2+8x+12} $ to $ \dfrac{9x+9}{x+6} $. Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{x+2}$. $$ \begin{aligned} \frac{9x^2+27x+18}{x^2+8x+12} & =\frac{ \left( 9x+9 \right) \cdot \color{blue}{ \left( x+2 \right) }}{ \left( x+6 \right) \cdot \color{blue}{ \left( x+2 \right) }} = \\[1ex] &= \frac{9x+9}{x+6} \end{aligned} $$ |
| ② | Simplify $ \dfrac{x^2+7x+6}{x^2+7x+6} $ to $ 1$. Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{x^2+7x+6}$. $$ \begin{aligned} \frac{x^2+7x+6}{x^2+7x+6} & =\frac{ 1 \cdot \color{blue}{ \left( x^2+7x+6 \right) }}{ 1 \cdot \color{blue}{ \left( x^2+7x+6 \right) }} = \\[1ex] &= \frac{1}{1} =1 \end{aligned} $$ |
| ③ | Divide $ \dfrac{x+6}{x+1} $ by $ 1 $ to get $ \dfrac{ x+6 }{ x+1 } $. 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. $$ \begin{aligned} \frac{ \frac{x+6}{x+1} }{1} & \xlongequal{\text{Step 1}} \frac{x+6}{x+1} \cdot \frac{\color{blue}{1}}{\color{blue}{1}} \xlongequal{\text{Step 2}} \frac{ \left( x+6 \right) \cdot 1 }{ \left( x+1 \right) \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x+6 }{ x+1 } \end{aligned} $$ |
| ④ | Multiply $ \dfrac{9x+9}{x+6} $ by $ \dfrac{x+6}{x+1} $ to get $ 9$. Step 1: Cancel $ \color{red}{ x+6 } $ in first and second fraction. Step 2: Factor numerators and denominators. Step 3: Cancel common factors. Step 4: Multiply numerators and denominators. $$ \begin{aligned} \frac{9x+9}{x+6} \cdot \frac{x+6}{x+1} & \xlongequal{\text{Step 1}} \frac{9x+9}{\color{red}{1}} \cdot \frac{\color{red}{1}}{x+1} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 9 \cdot \color{blue}{ \left( x+1 \right) } }{ 1 } \cdot \frac{ 1 }{ 1 \cdot \color{blue}{ \left( x+1 \right) } } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 9 }{ 1 } \cdot \frac{ 1 }{ 1 } \xlongequal{\text{Step 4}} \frac{ 9 }{ 1 } =9 \end{aligned} $$ |