Sure, let's tackle these problems step by step.
### Problem 3: [tex]\((2a + b)^3 - (2a - b)^3\)[/tex]
We are given the expression [tex]\((2a + b)^3 - (2a - b)^3\)[/tex]. To solve this, we can use the identity for the difference of cubes:
[tex]\[
x^3 - y^3 = (x - y)(x^2 + xy + y^2)
\][/tex]
Here, [tex]\(x = 2a + b\)[/tex] and [tex]\(y = 2a - b\)[/tex].
First, let's compute [tex]\(x - y\)[/tex]:
[tex]\[
x - y = (2a + b) - (2a - b) = 2a + b - 2a + b = 2b
\][/tex]
Next, we need to compute [tex]\(x^2 + xy + y^2\)[/tex]:
1. Compute [tex]\(x^2 = (2a + b)^2\)[/tex]:
[tex]\[
(2a + b)^2 = 4a^2 + 4ab + b^2
\][/tex]
2. Compute [tex]\(y^2 = (2a - b)^2\)[/tex]:
[tex]\[
(2a - b)^2 = 4a^2 - 4ab + b^2
\][/tex]
3. Compute [tex]\(xy = (2a + b)(2a - b)\)[/tex]:
[tex]\[
(2a + b)(2a - b) = 4a^2 - b^2
\][/tex]
Now, let's add these results together:
[tex]\[
x^2 + xy + y^2 = (4a^2 + 4ab + b^2) + (4a^2 - b^2) + (4a^2 - 4ab + b^2)
\][/tex]
Combine the like terms:
[tex]\[
4a^2 + 4ab + b^2 + 4a^2 - b^2 + 4a^2 - 4ab + b^2 = 12a^2 + b^2
\][/tex]
So, the expression becomes:
[tex]\[
(2a + b)^3 - (2a - b)^3 = 2b(12a^2 + b^2)
\][/tex]
Therefore, the simplified form of the expression is:
[tex]\[
24ab^2 + 2b^3
\][/tex]
### Problem 4a: [tex]\((198)^3\)[/tex]
To find [tex]\((198)^3\)[/tex], we calculate the cube of 198. The result is:
[tex]\[
198^3 = 198 \times 198 \times 198 = 7,762,392
\][/tex]
Therefore, the answers are:
- For problem 3: [tex]\(24ab^2 + 2b^3\)[/tex]
- For problem 4(a): [tex]\(7,762,392\)[/tex]
I hope this helps in understanding the procedures for solving these equations! If you have any more questions, feel free to ask.
