Question
Answer
$$\dfrac{(x+1)^2}{(x+1)^3}=\dfrac{1}{x+1}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{(x+1)^2}{(x+1)^3}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{x^2+2x+1}{x^3+3x^2+3x+1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{1}{x+1}\end{aligned} $$ | |
| ① | Find $ \left(x+1\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ x } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(x+1\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 1 + \color{red}{1^2} = x^2+2x+1\end{aligned} $$ |
| ② | Find $ \left(x+1\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = x $ and $ B = 1 $. $$ \left(x+1\right)^3 = x^3+3 \cdot x^2 \cdot 1 + 3 \cdot x \cdot 1^2+1^3 = x^3+3x^2+3x+1 $$ |
| ③ | Simplify $ \dfrac{x^2+2x+1}{x^3+3x^2+3x+1} $ to $ \dfrac{1}{x+1} $. Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{x^2+2x+1}$. $$ \begin{aligned} \frac{x^2+2x+1}{x^3+3x^2+3x+1} & =\frac{ 1 \cdot \color{blue}{ \left( x^2+2x+1 \right) }}{ \left( x+1 \right) \cdot \color{blue}{ \left( x^2+2x+1 \right) }} = \\[1ex] &= \frac{1}{x+1} \end{aligned} $$ |
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