I can help you solve the problems in the image, focusing on simplifying the algebraic expressions. Let's go through the problems you highlighted, one by one:

Problem 1:

$\frac{m^4 p^2}{m^2 p}$ Simplify the powers of $m$ and $p$: $= m^{4-2} p^{2-1} = m^2 p$

Problem 2:

$\frac{p^{12} r}{p^7 r}$ Simplify the powers of $p$ (since $r$ cancels out): $= p^{12-7} = p^5$

Problem 3:

$\frac{c^4 d^3 f^3}{c^2 d^4 f^2}$ Simplify the powers of $c$, $d$, and $f$: $= c^{4-2} d^{3-4} f^{3-2} = c^2 d^{-1} f = \frac{c^2 f}{d}$

Problem 4:

$\left( \frac{3xy^4}{5z^2} \right)^2$ Raise both the numerator and denominator to the power of 2: $= \frac{(3xy^4)^2}{(5z^2)^2} = \frac{9x^2 y^8}{25z^4}$

Problem 5:

$\left( \frac{p^2 t^7}{10} \right)^3$ Raise both the numerator and denominator to the power of 3: $= \frac{(p^2 t^7)^3}{10^3} = \frac{p^6 t^{21}}{1000}$

Would you like to continue with the other problems or need further details on these? Here are some related questions:

1. How do you simplify expressions involving negative exponents?
2. What happens when variables in the denominator cancel out entirely?
3. How do fractional exponents work in simplification problems?
4. How would you approach a similar problem involving fractions raised to negative powers?
5. What are the rules for simplifying when dealing with multiple variables raised to powers?

Tip: When simplifying, remember to subtract exponents when dividing like bases and to distribute powers across both the numerator and denominator.