Here are some moderately challenging exponent simplification problems along with step-by-step answers:

Problem 1

Simplify:
$\frac{(x^3 \cdot y^4)^2}{x^4 \cdot y^3}$

Solution:

1. Start by applying the power rule $(a^m)^n = a^{m \cdot n}$: $(x^3 \cdot y^4)^2 = x^{3 \cdot 2} \cdot y^{4 \cdot 2} = x^6 \cdot y^8$

2. Substitute back into the expression: $\frac{x^6 \cdot y^8}{x^4 \cdot y^3}$

3. Apply the division rule $a^m / a^n = a^{m - n}$: $= x^{6 - 4} \cdot y^{8 - 3} = x^2 \cdot y^5$

Answer:
$x^2 \cdot y^5$

---

Problem 2

Simplify:
$(3x^2 y^{-1})^3 \cdot (x^3 y^4)^{-1}$

Solution:

1. Start with the first part $(3x^2 y^{-1})^3$ and apply the power rule: $3^3 \cdot x^{2 \cdot 3} \cdot y^{-1 \cdot 3} = 27x^6 y^{-3}$

2. Now simplify the second part $(x^3 y^4)^{-1}$ by applying the power rule: $x^{3 \cdot -1} \cdot y^{4 \cdot -1} = x^{-3} y^{-4}$

3. Combine both parts: $27x^6 y^{-3} \cdot x^{-3} y^{-4}$

4. Apply the product rule $a^m \cdot a^n = a^{m + n}$: $= 27x^{6 + (-3)} y^{-3 + (-4)} = 27x^3 y^{-7}$

5. Write the final answer with positive exponents: $= \frac{27x^3}{y^7}$

Answer:
$\frac{27x^3}{y^7}$

---

Problem 3

Simplify:
$\left(\frac{x^2}{y^{-3}}\right)^4 \cdot y^5$

Solution:

1. Start by applying the power rule to the fraction $\left(\frac{x^2}{y^{-3}}\right)^4$: $= \frac{x^{2 \cdot 4}}{y^{-3 \cdot 4}} = \frac{x^8}{y^{-12}}$

2. Rewrite the fraction with positive exponents: $= x^8 \cdot y^{12}$

3. Multiply by $y^5$: $= x^8 \cdot y^{12 + 5} = x^8 \cdot y^{17}$

Answer:
$x^8 \cdot y^{17}$

---

Problem 4

Simplify:
$\frac{a^5 \cdot b^{-2}}{a^{-3} \cdot b^4}$

Solution:

1. Apply the division rule to each part of the fraction: $= a^{5 - (-3)} \cdot b^{-2 - 4}$

2. Simplify the exponents: $= a^{5 + 3} \cdot b^{-2 - 4} = a^8 \cdot b^{-6}$

3. Rewrite with positive exponents: $= \frac{a^8}{b^6}$

Answer:
$\frac{a^8}{b^6}$

---

Problem 5

Simplify:
$(x^{-2} \cdot y^3)^{-2} \cdot x^5 \cdot y^{-4}$

Solution:

1. Begin by simplifying $(x^{-2} \cdot y^3)^{-2}$ using the power rule: $= x^{-2 \cdot -2} \cdot y^{3 \cdot -2} = x^4 \cdot y^{-6}$

2. Substitute back into the expression: $x^4 \cdot y^{-6} \cdot x^5 \cdot y^{-4}$

3. Apply the product rule for each variable: $= x^{4 + 5} \cdot y^{-6 + (-4)} = x^9 \cdot y^{-10}$

4. Rewrite with positive exponents: $= \frac{x^9}{y^{10}}$

Answer:
$\frac{x^9}{y^{10}}$

---

Let me know if you need any additional explanations on these solutions.

---

Related Questions

1. Simplify: $\frac{(2a^3 b^{-2})^2}{a^4 b^{-3}}$
2. Simplify: $(5x^{-2} y^4)^{-3} \cdot (x^3 y)^{-2}$
3. Simplify: $\frac{(x^4 y^{-3})^2}{x^5 y^4}$
4. Simplify: $\left(\frac{3x^2}{y^{-3}}\right)^{-2} \cdot y^6$
5. Simplify: $\frac{(a^{-1} b^2)^{-3}}{a^4 b^{-3}}$

---

Tip

When simplifying exponents, remember that negative exponents indicate reciprocals (e.g., $a^{-n} = \frac{1}{a^n}$), and be cautious to keep signs of exponents correct during each step.