Let's solve part (f) of the problem visible in the image step by step. The expression provided is:

$\left(\frac{p^5}{q^4}\right)^2 \times \left(\frac{p^4}{q^5}\right)^{-3}$

Step-by-Step Solution:

1. Apply the power to each fraction: $\left(\frac{p^5}{q^4}\right)^2 = \frac{(p^5)^2}{(q^4)^2} = \frac{p^{10}}{q^8}$ $\left(\frac{p^4}{q^5}\right)^{-3} = \frac{(p^4)^{-3}}{(q^5)^{-3}} = \frac{p^{-12}}{q^{-15}}$ (Here, using the rule $(a^m)^n = a^{m \cdot n}$ and applying the negative exponent rule $(a/b)^{-n} = (b/a)^n$).

2. Multiply the two expressions: $\frac{p^{10}}{q^8} \times \frac{p^{-12}}{q^{-15}} = \frac{p^{10} \cdot p^{-12}}{q^8 \cdot q^{-15}}$

3. Simplify the powers of $p$ and $q$:

   • For $p$: $p^{10} \times p^{-12} = p^{10 + (-12)} = p^{-2}$
   • For $q$: $q^8 \times q^{-15} = q^{8 + (-15)} = q^{-7}$

4. Final simplified expression: $\frac{q^7}{p^2}$

The correct simplified result is:

$\frac{q^7}{p^2}$

Let me know if you would like details for the next part (h) or have any questions!

Here are some related questions:

1. How do you simplify exponents in a fraction when both the numerator and denominator are raised to powers?
2. What is the rule for multiplying terms with the same base but different exponents?
3. How does a negative exponent affect a fraction?
4. What is the difference between applying exponents inside and outside a fraction?
5. What happens when both the base and exponent are negative?

Tip: Remember that a negative exponent flips the base to the reciprocal (i.e., $a^{-n} = \frac{1}{a^n}$).