Certainly! Let's go through each part of the problem and find the solutions step-by-step.
### Part a: [tex]\((2x)^3\)[/tex]
To expand [tex]\((2x)^3\)[/tex], we need to cube the expression [tex]\(2x\)[/tex]:
[tex]\[
(2x)^3 = 2^3 \cdot x^3 = 8x^3
\][/tex]
So, the expanded form of [tex]\((2x)^3\)[/tex] is [tex]\(8x^3\)[/tex].
### Part b: [tex]\((3x - 2)^2\)[/tex]
To expand [tex]\((3x - 2)^2\)[/tex], we use the binomial theorem:
[tex]\[
(3x - 2)^2 = (3x - 2) \cdot (3x - 2)
\][/tex]
Applying the distributive property (FOIL method):
[tex]\[
(3x - 2)(3x - 2) = 3x \cdot 3x + 3x \cdot (-2) + (-2) \cdot 3x + (-2) \cdot (-2)
\][/tex]
[tex]\[
= 9x^2 - 6x - 6x + 4
\][/tex]
[tex]\[
= 9x^2 - 12x + 4
\][/tex]
So, the expanded form of [tex]\((3x - 2)^2\)[/tex] is [tex]\(9x^2 - 12x + 4\)[/tex].
### Part c: [tex]\((3x)^4\)[/tex]
To expand [tex]\((3x)^4\)[/tex], we need to raise the expression [tex]\(3x\)[/tex] to the fourth power:
[tex]\[
(3x)^4 = 3^4 \cdot x^4 = 81x^4
\][/tex]
So, the expanded form of [tex]\((3x)^4\)[/tex] is [tex]\(81x^4\)[/tex].
### Part d: [tex]\((3x)^{-3}\)[/tex]
To expand [tex]\((3x)^{-3}\)[/tex], we need to raise the expression [tex]\(3x\)[/tex] to the power of [tex]\(-3\)[/tex]:
[tex]\[
(3x)^{-3} = \frac{1}{(3x)^3} = \frac{1}{3^3 \cdot x^3} = \frac{1}{27x^3}
\][/tex]
So, the expanded form of [tex]\((3x)^{-3}\)[/tex] is [tex]\(\frac{1}{27x^3}\)[/tex].
### Summary of Results
a. [tex]\((2x)^3 = 8x^3\)[/tex]
b. [tex]\((3x - 2)^2 = 9x^2 - 12x + 4\)[/tex]
c. [tex]\((3x)^4 = 81x^4\)[/tex]
d. [tex]\((3x)^{-3} = \frac{1}{27x^3}\)[/tex]
These are the expanded forms of the given expressions.
