Sure, let's address each problem step-by-step.
### Problem 65: Combine like terms if any.
[tex]$65. \quad 4m^3y + 5x^2m - 8xmy + 7x^2m^2y^2$[/tex]
Inspecting the terms:
- [tex]\(4m^3y\)[/tex]
- [tex]\(5x^2m\)[/tex]
- [tex]\(-8xmy\)[/tex]
- [tex]\(7x^2m^2y^2\)[/tex]
We see that none of these terms are like terms because they each have different variables and/or exponents. Therefore, the expression is already simplified as:
[tex]$ 4m^3y + 5x^2m - 8xmy + 7x^2m^2y^2 $[/tex]
### Problem 66: Combine like terms.
[tex]$66. \quad -6t^2w + 12tw^2 - 3t^2w + 6tw$[/tex]
Combine like terms:
- The terms [tex]\(-6t^2w\)[/tex] and [tex]\(-3t^2w\)[/tex] are like terms.
- The terms [tex]\(12tw^2\)[/tex] and [tex]\(6tw\)[/tex] are not like terms with any other.
Calculate:
[tex]\[
-6t^2w - 3t^2w = (-6 - 3)t^2w = -9t^2w
\][/tex]
So the expression simplifies to:
[tex]$ -9t^2w + 12tw^2 + 6tw $[/tex]
### Problem 67: Combine like terms.
[tex]$67. \quad -\frac{1}{2}tz^2 + \frac{1}{4}tz^3 - \frac{1}{8}tz^4$[/tex]
Inspecting the terms:
- [tex]\(-\frac{1}{2}tz^2\)[/tex]
- [tex]\(\frac{1}{4}tz^3\)[/tex]
- [tex]\(-\frac{1}{8}tz^4\)[/tex]
We see that none of these terms are like terms because they each have different exponents of [tex]\(z\)[/tex]. Therefore, the expression is already simplified as:
[tex]$ -\frac{1}{2}tz^2 + \frac{1}{4}tz^3 - \frac{1}{8}tz^4 $[/tex]
Thus, here are the solutions:
1. For Problem 65: [tex]\(4m^3y + 5x^2m - 8xmy + 7x^2m^2y^2\)[/tex]
2. For Problem 66: [tex]\(-9t^2w + 12tw^2 + 6tw\)[/tex]
3. For Problem 67: [tex]\(-\frac{1}{2}tz^2 + \frac{1}{4}tz^3 - \frac{1}{8}tz^4\)[/tex]
