Sure, let's find the cubes of the given expressions one by one.
### Part (a): Cube of [tex]\((a + 4)\)[/tex]
To find the cube of the expression [tex]\((a + 4)\)[/tex], we use the binomial expansion formula for cubes:
[tex]\[
(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3
\][/tex]
Here, [tex]\(x = a\)[/tex] and [tex]\(y = 4\)[/tex].
Substituting these values into the formula:
[tex]\[
(a + 4)^3 = a^3 + 3(a^2)(4) + 3(a)(4^2) + 4^3
\][/tex]
[tex]\[
(a + 4)^3 = a^3 + 12a^2 + 48a + 64
\][/tex]
### Part (b): Cube of [tex]\((a - 5)\)[/tex]
Using the same binomial expansion formula, [tex]\((x + y)^3\)[/tex], for [tex]\((a - 5)\)[/tex]:
[tex]\[
x = a, \quad y = -5
\][/tex]
Substituting these values:
[tex]\[
(a - 5)^3 = a^3 + 3(a^2)(-5) + 3(a)(-5^2) + (-5)^3
\][/tex]
[tex]\[
(a - 5)^3 = a^3 - 15a^2 + 75a - 125
\][/tex]
### Part (e): Cube of [tex]\((2a + b)\)[/tex]
Now, we'll cube [tex]\((2a + b)\)[/tex]:
[tex]\[
x = 2a, \quad y = b
\][/tex]
Substituting:
[tex]\[
(2a + b)^3 = (2a)^3 + 3(2a)^2(b) + 3(2a)(b^2) + b^3
\][/tex]
[tex]\[
= 8a^3 + 12a^2b + 6ab^2 + b^3
\][/tex]
### Part (f): Cube of [tex]\((3a - 2)\)[/tex]
Finally, we cube [tex]\((3a - 2)\)[/tex]:
[tex]\[
x = 3a, \quad y = -2
\][/tex]
Substituting:
[tex]\[
(3a - 2)^3 = (3a)^3 + 3(3a)^2(-2) + 3(3a)(-2)^2 + (-2)^3
\][/tex]
[tex]\[
= 27a^3 - 54a^2 + 36a - 8
\][/tex]
### Summary
The cubes of the given expressions are:
1. [tex]\((a + 4)^3\)[/tex] is:
[tex]\[
a^3 + 12a^2 + 48a + 64
\][/tex]
2. [tex]\((a - 5)^3\)[/tex] is:
[tex]\[
a^3 - 15a^2 + 75a - 125
\][/tex]
3. [tex]\((2a + b)^3\)[/tex] is:
[tex]\[
8a^3 + 12a^2b + 6ab^2 + b^3
\][/tex]
4. [tex]\((3a - 2)^3\)[/tex] is:
[tex]\[
27a^3 - 54a^2 + 36a - 8
\][/tex]
