The problem is to simplify expressions with negative exponents. There are 12 expressions in total that need to be simplified. I will solve each of these problems.

1. Problem Description

The problem is to simplify expressions with negative exponents. There are 12 expressions in total that need to be simplified. I will solve each of these problems.

2. Solution Steps

Problem 1: $3^{-2}$

$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$

Problem 2: $3^{-4} \cdot 3^{5}$

Using the rule $a^m \cdot a^n = a^{m+n}$:

$3^{-4} \cdot 3^{5} = 3^{-4+5} = 3^{1} = 3$

Problem 3: $\frac{3^{1}}{3^{-2}}$

Using the rule $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{3^{1}}{3^{-2}} = 3^{1 - (-2)} = 3^{1+2} = 3^3 = 27$

Problem 4: $a^{-4}$

$a^{-4} = \frac{1}{a^4}$

Problem 5: $\frac{4b^{-4}}{\frac{4}{b}}$

$\frac{4b^{-4}}{\frac{4}{b}} = \frac{4 \cdot \frac{1}{b^4}}{\frac{4}{b}} = \frac{\frac{4}{b^4}}{\frac{4}{b}} = \frac{4}{b^4} \cdot \frac{b}{4} = \frac{4b}{4b^4} = \frac{1}{b^3}$

Problem 6: $\frac{1c}{4c^{-2}}$

$\frac{c}{4c^{-2}} = \frac{1}{4} \cdot \frac{c}{c^{-2}} = \frac{1}{4} c^{1 - (-2)} = \frac{1}{4} c^{1+2} = \frac{1}{4}c^3 = \frac{c^3}{4}$

Problem 7: $(5d^{2}) \cdot (2d^{-5})$

$(5d^{2}) \cdot (2d^{-5}) = 5 \cdot 2 \cdot d^{2} \cdot d^{-5} = 10 \cdot d^{2+(-5)} = 10 d^{-3} = \frac{10}{d^3}$

Problem 8: $\frac{9f^4}{3f^{-2}}$

$\frac{9f^4}{3f^{-2}} = \frac{9}{3} \cdot \frac{f^4}{f^{-2}} = 3 \cdot f^{4-(-2)} = 3f^{4+2} = 3f^6$

Problem 9: $(g^{-2})^{-2}$

Using the rule $(a^m)^n = a^{m \cdot n}$:

$(g^{-2})^{-2} = g^{(-2) \cdot (-2)} = g^{4}$

Problem 10: $(2h^{-3})^{2}$

$(2h^{-3})^{2} = 2^2 \cdot (h^{-3})^2 = 4 \cdot h^{-3 \cdot 2} = 4 h^{-6} = \frac{4}{h^6}$

Problem 11: $2jk^{-5}$

$2jk^{-5} = 2j \cdot \frac{1}{k^5} = \frac{2j}{k^5}$

Problem 12: $(\frac{15m}{5})^{-3}$

$(\frac{15m}{5})^{-3} = (3m)^{-3} = 3^{-3} \cdot m^{-3} = \frac{1}{3^3} \cdot \frac{1}{m^3} = \frac{1}{27} \cdot \frac{1}{m^3} = \frac{1}{27m^3}$

3. Final Answer

1. $3^{-2} = \frac{1}{9}$

2. $3^{-4} \cdot 3^{5} = 3$

3. $\frac{3^{1}}{3^{-2}} = 27$

4. $a^{-4} = \frac{1}{a^4}$

5. $\frac{4b^{-4}}{\frac{4}{b}} = \frac{1}{b^3}$

6. $\frac{1c}{4c^{-2}} = \frac{c^3}{4}$

7. $(5d^{2}) \cdot (2d^{-5}) = \frac{10}{d^3}$

8. $\frac{9f^4}{3f^{-2}} = 3f^6$

9. $(g^{-2})^{-2} = g^{4}$

1

0. $(2h^{-3})^{2} = \frac{4}{h^6}$

1

1. $2jk^{-5} = \frac{2j}{k^5}$

1

2. $(\frac{15m}{5})^{-3} = \frac{1}{27m^3}$