Sure! Let's expand each of the given expressions step-by-step.
### (a) [tex]\((a + 3)^3\)[/tex]:
To expand [tex]\((a + 3)^3\)[/tex], we use the binomial theorem, which states that [tex]\((x + y)^3\)[/tex] can be expanded as:
[tex]\[ (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3 \][/tex]
Here, [tex]\(x = a\)[/tex] and [tex]\(y = 3\)[/tex]. Applying this, we get:
[tex]\[ (a + 3)^3 = a^3 + 3(a^2)(3) + 3(a)(3^2) + 3^3 \][/tex]
[tex]\[ = a^3 + 9a^2 + 27a + 27 \][/tex]
So, the expanded form is:
[tex]\[ (a + 3)^3 = a^3 + 9a^2 + 27a + 27 \][/tex]
### (b) [tex]\((a - 4)^3\)[/tex]:
Similarly, using the binomial theorem for [tex]\((a - 4)^3\)[/tex], where [tex]\(x = a\)[/tex] and [tex]\(y = -4\)[/tex], we get:
[tex]\[ (a - 4)^3 = a^3 + 3(a^2)(-4) + 3(a)(-4)^2 + (-4)^3 \][/tex]
[tex]\[ = a^3 - 12a^2 + 48a - 64 \][/tex]
So, the expanded form is:
[tex]\[ (a - 4)^3 = a^3 - 12a^2 + 48a - 64 \][/tex]
### (e) [tex]\((2a - 3)^3\)[/tex]:
Again using the binomial theorem for [tex]\((2a - 3)^3\)[/tex], where [tex]\(x = 2a\)[/tex] and [tex]\(y = -3\)[/tex], we have:
[tex]\[ (2a - 3)^3 = (2a)^3 + 3(2a)^2(-3) + 3(2a)(-3)^2 + (-3)^3 \][/tex]
[tex]\[ = 8a^3 + 3(4a^2)(-3) + 3(2a)(9) + (-27) \][/tex]
[tex]\[ = 8a^3 - 36a^2 + 54a - 27 \][/tex]
So, the expanded form is:
[tex]\[ (2a - 3)^3 = 8a^3 - 36a^2 + 54a - 27 \][/tex]
### (f) [tex]\((3a + 2b)^3\)[/tex]:
Finally, using the binomial theorem for [tex]\((3a + 2b)^3\)[/tex], where [tex]\(x = 3a\)[/tex] and [tex]\(y = 2b\)[/tex], we get:
[tex]\[ (3a + 2b)^3 = (3a)^3 + 3(3a)^2(2b) + 3(3a)(2b)^2 + (2b)^3 \][/tex]
[tex]\[ = 27a^3 + 3(9a^2)(2b) + 3(3a)(4b^2) + 8b^3 \][/tex]
[tex]\[ = 27a^3 + 54a^2b + 36ab^2 + 8b^3 \][/tex]
So, the expanded form is:
[tex]\[ (3a + 2b)^3 = 27a^3 + 54a^2b + 36ab^2 + 8b^3 \][/tex]
Therefore, the expanded forms of the given expressions are:
[tex]\[ (a) \; (a + 3)^3 = a^3 + 9a^2 + 27a + 27 \][/tex]
[tex]\[ (b) \; (a - 4)^3 = a^3 - 12a^2 + 48a - 64 \][/tex]
[tex]\[ (e) \; (2a - 3)^3 = 8a^3 - 36a^2 + 54a - 27 \][/tex]
[tex]\[ (f) \; (3a + 2b)^3 = 27a^3 + 54a^2b + 36ab^2 + 8b^3 \][/tex]
