Let's solve the expression $3^4 \div 3^9$ and identify which of the given options are correct.

Step 1: Apply the Law of Exponents

When dividing powers with the same base, we subtract the exponents: $3^4 \div 3^9 = 3^{4 - 9} = 3^{-5}$

Step 2: Simplify Further

$3^{-5} = \frac{1}{3^5}$

Step 3: Compare with the Given Options

• $3^{13}$: Incorrect.
• $3^5$: Incorrect.
• $\frac{1}{243}$: Correct. Since $3^5 = 243$, $\frac{1}{3^5} = \frac{1}{243}$.
• $\frac{1}{3^5}$: Correct. This directly matches the simplified expression.
• $1^5$: Incorrect.

Correct Answers

• $\frac{1}{243}$
• $\frac{1}{3^5}$

Would you like further details or have any questions?

Here are 5 related questions:

1. What is the rule for multiplying powers with the same base?
2. How do you simplify expressions with negative exponents?
3. Can you simplify $\frac{1}{3^{-3}}$?
4. What is the value of $3^5$?
5. How would you simplify $2^4 \div 2^6$?

Tip: When dividing powers with the same base, always subtract the exponents.