To solve the expression [tex]\(\left(4 a^2 - 3 b^3\right)^2\)[/tex], we need to expand it using the binomial theorem or by carefully applying distributive properties. Here's a step-by-step explanation:
1. Express the square: Write the expression as a product of two identical terms.
[tex]\[
\left(4 a^2 - 3 b^3\right)^2 = \left(4 a^2 - 3 b^3\right) \cdot \left(4 a^2 - 3 b^3\right)
\][/tex]
2. Distribute the terms:
[tex]\[
(4 a^2 - 3 b^3) \cdot (4 a^2 - 3 b^3)
\][/tex]
We will apply the distributive property (also known as FOIL - First, Outer, Inner, Last) to expand this product:
- First: [tex]\(4 a^2 \cdot 4 a^2\)[/tex]
- Outer: [tex]\(4 a^2 \cdot (-3 b^3)\)[/tex]
- Inner: [tex]\((-3 b^3) \cdot 4 a^2\)[/tex]
- Last: [tex]\((-3 b^3) \cdot (-3 b^3)\)[/tex]
3. Calculate each term:
- First: [tex]\(4 a^2 \cdot 4 a^2 = 16 a^4\)[/tex]
- Outer: [tex]\(4 a^2 \cdot (-3 b^3) = -12 a^2 b^3\)[/tex]
- Inner: [tex]\((-3 b^3) \cdot 4 a^2 = -12 a^2 b^3\)[/tex]
- Last: [tex]\((-3 b^3) \cdot (-3 b^3) = 9 b^6\)[/tex]
4. Combine like terms:
[tex]\[
16 a^4 - 12 a^2 b^3 - 12 a^2 b^3 + 9 b^6
\][/tex]
Simplifying the middle terms:
[tex]\[
-12 a^2 b^3 - 12 a^2 b^3 = -24 a^2 b^3
\][/tex]
5. Write the final expanded expression:
[tex]\[
16 a^4 - 24 a^2 b^3 + 9 b^6
\][/tex]
Hence, the expanded form of [tex]\(\left(4 a^2 - 3 b^3\right)^2\)[/tex] is:
[tex]\[
16 a^4 - 24 a^2 b^3 + 9 b^6
\][/tex]
