Sure! Let's solve the expression step-by-step: [tex]\( 8(a+b)^3 - (a-b)^3 \)[/tex].
To find the expanded form of this expression, we will expand each term separately and then combine them.
1. Expand [tex]\( 8(a+b)^3 \)[/tex]:
The term [tex]\( (a+b)^3 \)[/tex] can be expanded using the binomial theorem:
[tex]\[
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\][/tex]
Multiply each term by 8:
[tex]\[
8(a+b)^3 = 8(a^3 + 3a^2b + 3ab^2 + b^3) = 8a^3 + 24a^2b + 24ab^2 + 8b^3
\][/tex]
2. Expand [tex]\( (a-b)^3 \)[/tex]:
Similarly, the term [tex]\( (a-b)^3 \)[/tex] can be expanded using the binomial theorem:
[tex]\[
(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
\][/tex]
3. Combine both expansions:
Now, we subtract the second expansion from the first:
[tex]\[
8a^3 + 24a^2b + 24ab^2 + 8b^3 - (a^3 - 3a^2b + 3ab^2 - b^3)
\][/tex]
Distribute the negative sign:
[tex]\[
= 8a^3 + 24a^2b + 24ab^2 + 8b^3 - a^3 + 3a^2b - 3ab^2 + b^3
\][/tex]
4. Combine like terms:
[tex]\[
= (8a^3 - a^3) + (24a^2b + 3a^2b) + (24ab^2 - 3ab^2) + (8b^3 + b^3)
= 7a^3 + 27a^2b + 21ab^2 + 9b^3
\][/tex]
Therefore, the expanded form of the expression [tex]\( 8(a+b)^3 - (a-b)^3 \)[/tex] is:
[tex]\[
\boxed{7a^3 + 27a^2b + 21ab^2 + 9b^3}
\][/tex]
