Math worksheet for practicing multiplying exponents by exponents, featuring 18 problems.

Let's solve the problem step by step.

The task is to multiply exponents, but it's important to understand what this means. When we multiply expressions with exponents, we need to consider whether the bases are the same or not.

Key Rules:

1. Same base: $ a^m \times a^n = a^{m+n} $
2. Different bases: $ a^m \times b^n $ → cannot be simplified further unless you compute the actual values.
3. Any number to the power of 0 is 1: $ a^0 = 1 $ (for $ a \neq 0 $)
4. $ a^1 = a $

We will evaluate each expression accordingly.

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Let’s go through each one:

#### Row 1:
1. $ 10^3 \times 10^2 = 10^{3+2} = 10^5 = 100,000 $
2. $ 11^3 \times 4^0 = 11^3 \times 1 = 11^3 = 1331 $
3. $ 20^1 \times 4^2 = 20 \times 16 = 320 $

#### Row 2:
4. $ 3^2 \times 3^3 = 3^{2+3} = 3^5 = 243 $
5. $ 2^4 \times 3^4 = 16 \times 81 = 1296 $
6. $ 5^4 \times 3^4 = 625 \times 81 = 50,625 $

#### Row 3:
7. $ 4^4 \times 3^2 = 256 \times 9 = 2,304 $
8. $ 1^4 \times 3^2 = 1 \times 9 = 9 $
9. $ 9^2 \times 3^2 = 81 \times 9 = 729 $

Note: $ 9^2 = (3^2)^2 = 3^4 $, so $ 9^2 \times 3^2 = 3^4 \times 3^2 = 3^6 = 729 $

#### Row 4:
10. $ 10^1 \times 10^3 = 10^{1+3} = 10^4 = 10,000 $
11. $ 6^3 \times 3^2 = 216 \times 9 = 1,944 $
12. $ 5^3 \times 7^2 = 125 \times 49 = 6,125 $

#### Row 5:
13. $ 2^4 \times 2^4 = 2^{4+4} = 2^8 = 256 $
14. $ 0^4 \times 9^2 = 0 \times 81 = 0 $
(Any non-zero number times 0 is 0)
15. $ 9^3 \times 3^1 = 729 \times 3 = 2,187 $

Note: $ 9^3 = (3^2)^3 = 3^6 $, so $ 3^6 \times 3^1 = 3^7 = 2,187 $

#### Row 6:
16. $ 6^3 \times 3^3 = 216 \times 27 = 5,832 $
17. $ 6^2 \times 4^3 = 36 \times 64 = 2,304 $
18. $ 5^2 \times 3^2 = 25 \times 9 = 225 $

#### Row 7:
19. $ 9^3 \times 3^5 = (3^2)^3 \times 3^5 = 3^6 \times 3^5 = 3^{11} = 177,147 $
20. $ 8^2 \times 3^3 = 64 \times 27 = 1,728 $
21. $ 2^4 \times 8^3 = 16 \times (2^3)^3 = 16 \times 2^9 = 2^4 \times 2^9 = 2^{13} = 8,192 $

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Final Answers:

| Problem | Answer |
|--------|--------|
| $ 10^3 \times 10^2 $ | $ 10^5 = 100,000 $ |
| $ 11^3 \times 4^0 $ | $ 1331 $ |
| $ 20^1 \times 4^2 $ | $ 320 $ |
| $ 3^2 \times 3^3 $ | $ 3^5 = 243 $ |
| $ 2^4 \times 3^4 $ | $ 16 \times 81 = 1296 $ |
| $ 5^4 \times 3^4 $ | $ 625 \times 81 = 50,625 $ |
| $ 4^4 \times 3^2 $ | $ 256 \times 9 = 2,304 $ |
| $ 1^4 \times 3^2 $ | $ 1 \times 9 = 9 $ |
| $ 9^2 \times 3^2 $ | $ 81 \times 9 = 729 $ |
| $ 10^1 \times 10^3 $ | $ 10^4 = 10,000 $ |
| $ 6^3 \times 3^2 $ | $ 216 \times 9 = 1,944 $ |
| $ 5^3 \times 7^2 $ | $ 125 \times 49 = 6,125 $ |
| $ 2^4 \times 2^4 $ | $ 2^8 = 256 $ |
| $ 0^4 \times 9^2 $ | $ 0 $ |
| $ 9^3 \times 3^1 $ | $ 729 \times 3 = 2,187 $ |
| $ 6^3 \times 3^3 $ | $ 216 \times 27 = 5,832 $ |
| $ 6^2 \times 4^3 $ | $ 36 \times 64 = 2,304 $ |
| $ 5^2 \times 3^2 $ | $ 25 \times 9 = 225 $ |
| $ 9^3 \times 3^5 $ | $ 3^{11} = 177,147 $ |
| $ 8^2 \times 3^3 $ | $ 64 \times 27 = 1,728 $ |
| $ 2^4 \times 8^3 $ | $ 2^{13} = 8,192 $ |

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Summary of Techniques Used:

- Same base: Add exponents.
- Different bases: Multiply the values directly.
- Zero exponent: $ a^0 = 1 $
- Zero base: $ 0^n = 0 $ for $ n > 0 $
- Powers of powers: $ (a^m)^n = a^{m \times n} $
- Simplify when possible using prime factorization (e.g., $ 8 = 2^3 $, $ 9 = 3^2 $)

Let me know if you'd like these answers in a printable format or explained further!