https://file.aiquickdraw.com/mathbot/file/172698959911870vkiena.jpg

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., \(a^{-n} = \frac{1}{a^n}\)).

Simplify the expression: (p^5 / q^4)^2 × (p^4 / q^5)^-3

Learn how to simplify complex expressions involving exponents and fractions with both positive and negative powers. The problem covers key concepts like power of a power, multiplying exponents, and handling negative exponents. Suitable for students in Grades 9-11.

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