Question
Answer
$$2(v+2x)^4=2v^4+16v^3x+48v^2x^2+64vx^3+32x^4$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}2(v+2x)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}2(1v^4+8v^3x+24v^2x^2+32vx^3+16x^4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}2v^4+16v^3x+48v^2x^2+64vx^3+32x^4\end{aligned} $$ | |
| ① | $$ (v+2x)^4 = (v+2x)^2 \cdot (v+2x)^2 $$ |
| ② | Find $ \left(v+2x\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ v } $ and $ B = \color{red}{ 2x }$. $$ \begin{aligned}\left(v+2x\right)^2 = \color{blue}{v^2} +2 \cdot v \cdot 2x + \color{red}{\left( 2x \right)^2} = v^2+4vx+4x^2\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{v^2+4vx+4x^2}\right) $ by each term in $ \left( v^2+4vx+4x^2\right) $. $$ \left( \color{blue}{v^2+4vx+4x^2}\right) \cdot \left( v^2+4vx+4x^2\right) = \\ = v^4+4v^3x+4v^2x^2+4v^3x+16v^2x^2+16vx^3+4v^2x^2+16vx^3+16x^4 $$ |
| ④ | Combine like terms: $$ v^4+ \color{blue}{4v^3x} + \color{red}{4v^2x^2} + \color{blue}{4v^3x} + \color{green}{16v^2x^2} + \color{orange}{16vx^3} + \color{green}{4v^2x^2} + \color{orange}{16vx^3} +16x^4 = \\ = v^4+ \color{blue}{8v^3x} + \color{green}{24v^2x^2} + \color{orange}{32vx^3} +16x^4 $$ |
| ⑤ | Multiply $ \color{blue}{2} $ by $ \left( v^4+8v^3x+24v^2x^2+32vx^3+16x^4\right) $ $$ \color{blue}{2} \cdot \left( v^4+8v^3x+24v^2x^2+32vx^3+16x^4\right) = 2v^4+16v^3x+48v^2x^2+64vx^3+32x^4 $$ |
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