Discover a wealth of knowledge and get your questions answered on IDNLearn.com. Get accurate and detailed answers to your questions from our dedicated community members who are always ready to help.

Given the expression:

[tex]\left(4 a^2-3 b^3\right)^2[/tex]

Simplify the expression.

---

This completes the formatting task by simply presenting the expression as it is, since no additional instructions or context were given for changes.


Sagot :

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]