Let's analyze the question and answer choices about properties of exponents based on your image.

Properties of Exponents Overview:

1. Product of Powers Rule:
   $a^m \cdot a^n = a^{m+n}$

2. Quotient of Powers Rule:
   $\frac{a^m}{a^n} = a^{m-n}$ (assuming $a \neq 0$)

3. Negative Exponent Rule:
   $a^{-n} = \frac{1}{a^n}$ for $a \neq 0$

Now let’s look at each option:

Analysis of Answer Choices:

1. $a^m \cdot a^n = a^{m - n}$

   • Incorrect. The correct rule for multiplying powers with the same base is $a^m \cdot a^n = a^{m + n}$. This option incorrectly uses subtraction instead of addition.

2. If $a$ is nonzero, and $n$ is a rational number, then $a^{-n} = \frac{1}{a^n}$

   • Correct. This is the negative exponent rule.

3. $\frac{a^m}{a^n} = a^{m - n}$

   • Correct. This is the quotient of powers rule.

4. If $a$ is nonzero, and $n$ is a rational number, then $a^{-n} = \frac{1}{a^{-n}}$

   • Incorrect. The correct expression should be $a^{-n} = \frac{1}{a^n}$. The option contains a mistake by placing a negative exponent in the denominator.

Correct Answer Choices:

• Choice 2: $a^{-n} = \frac{1}{a^n}$
• Choice 3: $\frac{a^m}{a^n} = a^{m - n}$

Let me know if this clarifies it or if you have additional questions!

Would you like more details or another breakdown? Here are five related questions:

1. What is the zero exponent rule?
2. How do fractional exponents relate to roots?
3. Can exponents be negative or zero for all real numbers?
4. What happens when an exponent is applied to a product, like $(ab)^n$?
5. How do exponent rules change with variables or algebraic expressions?

Tip: When in doubt with exponent rules, write out the terms explicitly to spot patterns.