Answer
$$(a+b)^3-(a-b)^3=6a^2b+2b^3$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(a+b)^3-(a-b)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}a^3+3a^2b+3ab^2+b^3-(1a^3-3a^2b+3ab^2-b^3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}a^3+3a^2b+3ab^2+b^3-a^3+3a^2b-3ab^2+b^3 \xlongequal{ } \\[1 em] & \xlongequal{ } \cancel{a^3}+3a^2b+ \cancel{3ab^2}+b^3 -\cancel{a^3}+3a^2b -\cancel{3ab^2}+b^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}6a^2b+2b^3\end{aligned} $$ | |
| ① | Find $ \left(a+b\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = a $ and $ B = b $. $$ \left(a+b\right)^3 = a^3+3 \cdot a^2 \cdot b + 3 \cdot a \cdot b^2+b^3 = a^3+3a^2b+3ab^2+b^3 $$Find $ \left(a-b\right)^3 $ using formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = a $ and $ B = b $. $$ \left(a-b\right)^3 = a^3-3 \cdot a^2 \cdot b + 3 \cdot a \cdot b^2-b^3 = a^3-3a^2b+3ab^2-b^3 $$ |
| ② | Remove the parentheses by changing the sign of each term within them. $$ - \left( a^3-3a^2b+3ab^2-b^3 \right) = -a^3+3a^2b-3ab^2+b^3 $$ |
| ③ | Combine like terms: $$ \, \color{blue}{ \cancel{a^3}} \,+ \color{green}{3a^2b} + \, \color{orange}{ \cancel{3ab^2}} \,+ \color{red}{b^3} \, \color{blue}{ -\cancel{a^3}} \,+ \color{green}{3a^2b} \, \color{orange}{ -\cancel{3ab^2}} \,+ \color{red}{b^3} = \color{green}{6a^2b} + \color{red}{2b^3} $$ |
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