Sure! To expand and simplify the given expression [tex]\((4a^2 - 3b^3)^2\)[/tex], we will use the algebraic identity [tex]\((x - y)^2 = x^2 - 2xy + y^2\)[/tex]. Here, we have [tex]\(x = 4a^2\)[/tex] and [tex]\(y = 3b^3\)[/tex].
Let's go through the steps:
1. Identify the expression and components:
[tex]\[
(4a^2 - 3b^3)^2
\][/tex]
Here, [tex]\(x = 4a^2\)[/tex] and [tex]\(y = 3b^3\)[/tex].
2. Apply the square of a binomial formula:
[tex]\[
(x - y)^2 = x^2 - 2xy + y^2
\][/tex]
3. Calculate each term:
- [tex]\(x^2 = (4a^2)^2 = 16a^4\)[/tex]
- [tex]\(y^2 = (3b^3)^2 = 9b^6\)[/tex]
- [tex]\(2xy = 2(4a^2)(3b^3) = 24a^2b^3\)[/tex]
4. Combine these results:
[tex]\[
(4a^2 - 3b^3)^2 = (4a^2)^2 - 2(4a^2)(3b^3) + (3b^3)^2
\][/tex]
Substituting the calculated terms in:
[tex]\[
(4a^2 - 3b^3)^2 = 16a^4 - 24a^2b^3 + 9b^6
\][/tex]
So, the expanded form of [tex]\((4a^2 - 3b^3)^2\)[/tex] is:
[tex]\[
16a^4 - 24a^2b^3 + 9b^6
\][/tex]
This is the simplified form of the given expression.
