Let's solve the problems one by one, breaking them into parts:

Part F: Negative Exponent Rule

Using the rule $a^{-n} = \frac{1}{a^n}$:

1. $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$
2. $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$
3. $3^{-7} = \frac{1}{3^7} = \frac{1}{2187}$
4. $10^{-4} = \frac{1}{10^4} = \frac{1}{10000}$
5. $3^{-5} = \frac{1}{3^5} = \frac{1}{243}$

Part G: Simplifying Expressions with Negative Exponents

1. $\frac{1}{2^{-4}} = 2^4 = 16$
2. $\frac{1}{5^{-3}} = 5^3 = 125$
3. $\frac{1}{10^{-2}} = 10^2 = 100$
4. $\frac{1}{3^{-1}} = 3^1 = 3$
5. $\frac{1}{6^{-2}} = 6^2 = 36$

Part H: Mixed Expressions with Positive, Negative, and Zero Exponents

1. $3^{-2} \times 3^3 = 3^{-2+3} = 3^1 = 3$
2. $4^0 \times 4^{-2} = 1 \times \frac{1}{4^2} = \frac{1}{16}$
3. $5^1 \times 5^2 = 5^{1+2} = 5^3 = 125$
4. $2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$
5. $10^0 \times 10^1 = 1 \times 10 = 10$

Part I: Division with Negative Exponents

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

1. $7^{-3} \div 7^5 = 7^{-3-5} = 7^{-8} = \frac{1}{7^8}$
2. $9^2 \div 9^4 = 9^{2-4} = 9^{-2} = \frac{1}{9^2} = \frac{1}{81}$
3. $8^1 \div 8^2 = 8^{1-2} = 8^{-1} = \frac{1}{8}$
4. $6^0 \div 6^2 = 6^{0-2} = 6^{-2} = \frac{1}{6^2} = \frac{1}{36}$
5. $2^3 \div 2^1 = 2^{3-1} = 2^2 = 4$

Bonus Question: Simplify $(5^{-3} \times 5^2) \div 5^0$

• First, simplify $5^{-3} \times 5^2 = 5^{-3+2} = 5^{-1} = \frac{1}{5}$
• Since $5^0 = 1$, we have $\frac{1}{5} \div 1 = \frac{1}{5}$

The final answer is $\frac{1}{5}$.

Would you like more details on any of these steps, or do you have specific questions about the rules used?

Related Questions:

1. What is the difference between negative and zero exponents?
2. How do fractional exponents relate to radicals?
3. How would you solve expressions with both multiplication and division of exponents?
4. Can negative exponents ever produce a negative result?
5. How does the power of a product rule apply to multiple bases with exponents?

Tip:

Always simplify exponents first before performing any other operations in an expression to avoid mistakes.