Exponents Worksheets with Answer Key - Free Printable

Let’s solve each problem using the Product Rule for Exponents.

The rule is simple:
When you multiply two powers with the same base, you add the exponents.
That means:
> \( a^m \times a^n = a^{m+n} \)

We’ll apply this to every problem. Let’s go one by one.

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Problem 1: \( 12^{12} \times 12^{15} \)
Same base (12), so add exponents:
\( 12 + 15 = 27 \) → Answer: \( 12^{27} \)

Problem 2: \( 14^{17} \times 14^{20} \)
Add exponents: \( 17 + 20 = 37 \) → Answer: \( 14^{37} \)

Problem 3: \( 1^{11} \times 1^{18} \)
Add exponents: \( 11 + 18 = 29 \) → Answer: \( 1^{29} \)
*(Note: Any power of 1 is still 1, but we write it as \( 1^{29} \) per instructions.)*

Problem 4: \( 15^{10} \times 15^{17} \)
Add exponents: \( 10 + 17 = 27 \) → Answer: \( 15^{27} \)

Problem 5: \( 8^4 \times 8^6 \)
Add exponents: \( 4 + 6 = 10 \) → Answer: \( 8^{10} \)

Problem 6: \( 4^6 \times 4^4 \)
Add exponents: \( 6 + 4 = 10 \) → Answer: \( 4^{10} \)

Problem 7: \( 3^{19} \times 3^{16} \)
Add exponents: \( 19 + 16 = 35 \) → Answer: \( 3^{35} \)

Problem 8: \( 6^9 \times 6^{12} \)
Add exponents: \( 9 + 12 = 21 \) → Answer: \( 6^{21} \)

Problem 9: \( 13^{20} \times 13^4 \)
Add exponents: \( 20 + 4 = 24 \) → Answer: \( 13^{24} \)

Problem 10: \( 2^{16} \times 2^{11} \)
Add exponents: \( 16 + 11 = 27 \) → Answer: \( 2^{27} \)

Problem 11: \( 14^{17} \times 14^{22} \)
Add exponents: \( 17 + 22 = 39 \) → Answer: \( 14^{39} \)

Problem 12: \( 19^{16} \times 19^{12} \)
Add exponents: \( 16 + 12 = 28 \) → Answer: \( 19^{28} \)

Problem 13: \( 10^0 \times 10^1 \)
Add exponents: \( 0 + 1 = 1 \) → Answer: \( 10^1 \)
*(Remember: \( 10^0 = 1 \), and \( 1 \times 10 = 10 \), which is \( 10^1 \).)*

Problem 14: \( 11^2 \times 11^8 \)
Add exponents: \( 2 + 8 = 10 \) → Answer: \( 11^{10} \)

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All problems solved correctly by adding exponents when bases are the same.

Final Answer:
1. \( 12^{27} \)
2. \( 14^{37} \)
3. \( 1^{29} \)
4. \( 15^{27} \)
5. \( 8^{10} \)
6. \( 4^{10} \)
7. \( 3^{35} \)
8. \( 6^{21} \)
9. \( 13^{24} \)
10. \( 2^{27} \)
11. \( 14^{39} \)
12. \( 19^{28} \)
13. \( 10^1 \)
14. \( 11^{10} \)