Certainly! Let's proceed to expand each expression step by step.
### (a) [tex]\((a + 3)^3\)[/tex]
To expand [tex]\((a + 3)^3\)[/tex], we use the binomial theorem:
[tex]\[
(a + 3)^3 = a^3 + 3 \cdot a^2 \cdot 3 + 3 \cdot a \cdot 3^2 + 3^3
\][/tex]
Simplifying:
[tex]\[
a^3 + 9a^2 + 27a + 27
\][/tex]
So, [tex]\((a + 3)^3 = a^3 + 9a^2 + 27a + 27\)[/tex].
### (b) [tex]\((a - 4)^3\)[/tex]
To expand [tex]\((a - 4)^3\)[/tex], we use the binomial theorem:
[tex]\[
(a - 4)^3 = a^3 + 3 \cdot a^2 \cdot (-4) + 3 \cdot a \cdot (-4)^2 + (-4)^3
\][/tex]
Simplifying:
[tex]\[
a^3 - 12a^2 + 48a - 64
\][/tex]
So, [tex]\((a - 4)^3 = a^3 - 12a^2 + 48a - 64\)[/tex].
### (c) [tex]\((x + 2)^3\)[/tex]
To expand [tex]\((x + 2)^3\)[/tex], we use the binomial theorem:
[tex]\[
(x + 2)^3 = x^3 + 3 \cdot x^2 \cdot 2 + 3 \cdot x \cdot 2^2 + 2^3
\][/tex]
Simplifying:
[tex]\[
x^3 + 6x^2 + 12x + 8
\][/tex]
So, [tex]\((x + 2)^3 = x^3 + 6x^2 + 12x + 8\)[/tex].
### (d) [tex]\((x - 5)^3\)[/tex]
To expand [tex]\((x - 5)^3\)[/tex], we use the binomial theorem:
[tex]\[
(x - 5)^3 = x^3 + 3 \cdot x^2 \cdot (-5) + 3 \cdot x \cdot (-5)^2 + (-5)^3
\][/tex]
Simplifying:
[tex]\[
x^3 - 15x^2 + 75x - 125
\][/tex]
So, [tex]\((x - 5)^3 = x^3 - 15x^2 + 75x - 125\)[/tex].
### (e) [tex]\((2a - 3)^3\)[/tex]
To expand [tex]\((2a - 3)^3\)[/tex], we use the binomial theorem:
[tex]\[
(2a - 3)^3 = (2a)^3 + 3 \cdot (2a)^2 \cdot (-3) + 3 \cdot (2a) \cdot (-3)^2 + (-3)^3
\][/tex]
Simplifying:
[tex]\[
8a^3 - 36a^2 + 54a - 27
\][/tex]
So, [tex]\((2a - 3)^3 = 8a^3 - 36a^2 + 54a - 27\)[/tex].
### (f) [tex]\((3a + 2b)^3\)[/tex]
To expand [tex]\((3a + 2b)^3\)[/tex], we use the binomial theorem:
[tex]\[
(3a + 2b)^3 = (3a)^3 + 3 \cdot (3a)^2 \cdot (2b) + 3 \cdot (3a) \cdot (2b)^2 + (2b)^3
\][/tex]
Simplifying:
[tex]\[
27a^3 + 54a^2b + 36ab^2 + 8b^3
\][/tex]
So, [tex]\((3a + 2b)^3 = 27a^3 + 54a^2b + 36ab^2 + 8b^3\)[/tex].
### (g) [tex]\((5x - y)^3\)[/tex]
To expand [tex]\((5x - y)^3\)[/tex], we use the binomial theorem:
[tex]\[
(5x - y)^3 = (5x)^3 + 3 \cdot (5x)^2 \cdot (-y) + 3 \cdot (5x) \cdot (-y)^2 + (-y)^3
\][/tex]
Simplifying:
[tex]\[
125x^3 - 75x^2y + 15xy^2 - y^3
\][/tex]
So, [tex]\((5x - y)^3 = 125x^3 - 75x^2y + 15xy^2 - y^3\)[/tex].
### (h) [tex]\((3m + 2n)^3\)[/tex]
To expand [tex]\((3m + 2n)^3\)[/tex], we use the binomial theorem:
[tex]\[
(3m + 2n)^3 = (3m)^3 + 3 \cdot (3m)^2 \cdot (2n) + 3 \cdot (3m) \cdot (2n)^2 + (2n)^3
\][/tex]
Simplifying:
[tex]\[
27m^3 + 54m^2n + 36mn^2 + 8n^3
\][/tex]
So, [tex]\((3m + 2n)^3 = 27m^3 + 54m^2n + 36mn^2 + 8n^3\)[/tex].
These are the expanded forms of the given expressions.
