Exponents Product Rule worksheet for practicing simplifying expressions using the product rule.

Let's solve this step by step using the Exponents Product Rule.

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🔷 Product Rule for Exponents:

When multiplying two powers with the same base, you add the exponents:

> $$
> a^m \times a^n = a^{m+n}
> $$

We will apply this rule to each problem on the worksheet.

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✔ Solving Each Problem:

#### 1. $ 12^{14} \times 12^2 $
- Same base (12), so add exponents:
- $ 12^{14+2} = 12^{16} $

#### 2. $ 8^5 \times 8^8 $
- $ 8^{5+8} = 8^{13} $

#### 3. $ 13^3 \times 13^3 $
- $ 13^{3+3} = 13^6 $

#### 4. $ 5^8 \times 5^1 $
- $ 5^{8+1} = 5^9 $

#### 5. $ 3^8 \times 3^3 $
- $ 3^{8+3} = 3^{11} $

#### 6. $ 8^{15} \times 8^1 $
- $ 8^{15+1} = 8^{16} $

#### 7. $ 6^3 \times 6^2 $
- $ 6^{3+2} = 6^5 $

#### 8. $ 2^7 \times 2^2 $
- $ 2^{7+2} = 2^9 $

#### 9. $ 16^9 \times 16^3 $
- $ 16^{9+3} = 16^{12} $

#### 10. $ 4^5 \times 4^6 $
- $ 4^{5+6} = 4^{11} $

#### 11. $ 10^6 \times 10^8 $
- $ 10^{6+8} = 10^{14} $

#### 12. $ 1^1 \times 1^5 $
- $ 1^{1+5} = 1^6 $
- Note: Any power of 1 is still 1, but we keep it as $ 1^6 $ per instruction.

#### 13. $ 14^3 \times 14^2 $
- $ 14^{3+2} = 14^5 $

#### 14. $ 30^{20} \times 30^5 $
- $ 30^{20+5} = 30^{25} $

#### 15. $ 3^9 \times 3^5 $
- $ 3^{9+5} = 3^{14} $

#### 16. $ 7^{11} \times 7^8 $
- $ 7^{11+8} = 7^{19} $

#### 17. $ 5^7 \times 5^3 $
- $ 5^{7+3} = 5^{10} $

#### 18. $ 8^{14} \times 8^9 $
- $ 8^{14+9} = 8^{23} $

#### 19. $ 11^{20} \times 11^{12} $
- $ 11^{20+12} = 11^{32} $

#### 20. $ 7^{11} \times 7^2 $
- $ 7^{11+2} = 7^{13} $

#### 21. $ 19^4 \times 19^3 $
- $ 19^{4+3} = 19^7 $

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✔ Final Answers (in single exponent form):

| Problem | Answer |
|--------|--------|
| $ 12^{14} \times 12^2 $ | $ 12^{16} $ |
| $ 8^5 \times 8^8 $ | $ 8^{13} $ |
| $ 13^3 \times 13^3 $ | $ 13^6 $ |
| $ 5^8 \times 5^1 $ | $ 5^9 $ |
| $ 3^8 \times 3^3 $ | $ 3^{11} $ |
| $ 8^{15} \times 8^1 $ | $ 8^{16} $ |
| $ 6^3 \times 6^2 $ | $ 6^5 $ |
| $ 2^7 \times 2^2 $ | $ 2^9 $ |
| $ 16^9 \times 16^3 $ | $ 16^{12} $ |
| $ 4^5 \times 4^6 $ | $ 4^{11} $ |
| $ 10^6 \times 10^8 $ | $ 10^{14} $ |
| $ 1^1 \times 1^5 $ | $ 1^6 $ |
| $ 14^3 \times 14^2 $ | $ 14^5 $ |
| $ 30^{20} \times 30^5 $ | $ 30^{25} $ |
| $ 3^9 \times 3^5 $ | $ 3^{14} $ |
| $ 7^{11} \times 7^8 $ | $ 7^{19} $ |
| $ 5^7 \times 5^3 $ | $ 5^{10} $ |
| $ 8^{14} \times 8^9 $ | $ 8^{23} $ |
| $ 11^{20} \times 11^{12} $ | $ 11^{32} $ |
| $ 7^{11} \times 7^2 $ | $ 7^{13} $ |
| $ 19^4 \times 19^3 $ | $ 19^7 $ |

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📝 Summary:

All problems follow the product rule: when multiplying like bases, add the exponents. This is a fundamental rule in algebra and simplifies expressions involving powers.

✔ Final Answer: All expressions are simplified to a single exponent as shown above.