Question

simplify:(a+b)3-(a-b)3-6b(a2-b2)

Answer 1

Have a look at this picture.

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Answer 2

$(a+b)^{3}-(a-b)^{3}-6b(a^{2}-b^{2})=8b^{3}$

Tips :

Before we solve the problem, we must know some algebraic identities:

• $(a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}$
• $(a-b)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}$

Step-by-step explanation:

Step 1. Write down the given expression

$\quad (a+b)^{3}-(a-b)^{3}-6b(a^{2}-b^{2})$

Step 2. Use algebraic identities for $(a+b)^{3}$ $(a-b)^{3}$

$\quad (a^{3}+3a^{2}b+3ab^{2}+b^{3})-(a^{3}-3a^{2}b+3ab^{2}-b^{3})-6b(a^{2}-b^{2})$

Step 3. Use $(+)(+)=(+)$, $(+)(-)=(-)$, $(-)(+)=(-)$ and $(-)(+)=(-)$ to open the braces and rearrange the terms

$\quad a^{3}+3a^{2}b+3ab^{2}+b^{3}-a^{3}+3a^{2}b-3ab^{2}+b^{3}-6a^{2}b+6b^{3}$

$=a^{3}-a^{3}+b^{3}+b^{3}+6b^{3}+3a^{2}b+3a^{2}b-6a^{2}b+3ab^{2}-3ab^{2}$

Step 4. Now calculate the terms using algebraic operations

$\quad 8b^{3}$

Thus, the required simplified form is $8b^{3}$.