Multiplying and Dividing Exponents worksheet with 12 problems for practice.

Let’s solve each problem step by step. We’ll use the rules of exponents:

- When multiplying powers with the same base, add the exponents:
\( a^m \times a^n = a^{m+n} \)

- When dividing powers with the same base, subtract the exponents:
\( \frac{a^m}{a^n} = a^{m-n} \)

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Problem 1: \( 4^5 \times 4^3 \times 4^{10} \)
Add exponents: \( 5 + 3 + 10 = 18 \)
→ \( 4^{18} \)

Problem 2: \( 2^3 \times 2^{-4} \times 2^6 \)
Add exponents: \( 3 + (-4) + 6 = 5 \)
→ \( 2^5 \)

Problem 3: \( 5^7 \times 5^{-8} \times 5^{10} \)
Add exponents: \( 7 + (-8) + 10 = 9 \)
→ \( 5^9 \)

Problem 4: \( 6^7 \times 6^2 \times 6^8 \)
Add exponents: \( 7 + 2 + 8 = 17 \)
→ \( 6^{17} \)

Problem 5: \( 9^6 \times 9^4 \times 9^4 \)
Add exponents: \( 6 + 4 + 4 = 14 \)
→ \( 9^{14} \)

Problem 6: \( 7^3 \times 7^5 \times 7^8 \)
Add exponents: \( 3 + 5 + 8 = 16 \)
→ \( 7^{16} \)

Problem 7: \( \frac{7^6}{7^3} \)
Subtract exponents: \( 6 - 3 = 3 \)
→ \( 7^3 \)

Problem 8: \( \frac{8^4}{8^8} \)
Subtract exponents: \( 4 - 8 = -4 \)
→ \( 8^{-4} \) (or you can write as \( \frac{1}{8^4} \), but since the problems don’t specify, we leave it as negative exponent unless told otherwise — however, looking at other answers, they seem to prefer positive exponents or simplified form. But in this case, since no instruction is given, and all others are written as single power, we keep as \( 8^{-4} \). Wait — let me check if any answer should be rewritten.)

Actually, looking at the pattern, for division, sometimes they expect positive exponents. Let’s see Problem 9: \( \frac{4^9}{4^5} = 4^4 \) → positive.

But Problem 8: \( 8^4 / 8^8 = 8^{-4} \). That’s correct mathematically. However, some curricula prefer writing without negative exponents. But since the original problems include negative exponents in multiplication (like problem 2 and 3), it’s okay to leave negative exponents.

Wait — actually, in problem 2 and 3, the negative exponents were part of the input, not the output. The outputs here are results. Let me double-check standard practice.

In most middle school worksheets like this, they accept negative exponents unless specified otherwise. But let’s look at problem 8 again: \( \frac{8^4}{8^8} = 8^{4-8} = 8^{-4} \). That’s fine.

However, I notice that in the worksheet, all answers are expected to be in exponential form, possibly with negative exponents allowed. So we’ll keep them as is.

But wait — let me verify with problem 10: \( \frac{5^9}{5^4} = 5^5 \) → positive.

So only when numerator exponent < denominator exponent do we get negative.

We’ll proceed correctly.

Problem 8: \( \frac{8^4}{8^8} = 8^{-4} \)

Problem 9: \( \frac{4^9}{4^5} = 4^{9-5} = 4^4 \)

Problem 10: \( \frac{5^9}{5^4} = 5^{9-4} = 5^5 \)

Problem 11: \( \frac{6^7}{6^2} = 6^{7-2} = 6^5 \)

Problem 12: \( \frac{9^6}{9^3} = 9^{6-3} = 9^3 \)

Now, let’s list all final answers clearly.

Final Answer:
1. \( 4^{18} \)
2. \( 2^5 \)
3. \( 5^9 \)
4. \( 6^{17} \)
5. \( 9^{14} \)
6. \( 7^{16} \)
7. \( 7^3 \)
8. \( 8^{-4} \)
9. \( 4^4 \)
10. \( 5^5 \)
11. \( 6^5 \)
12. \( 9^3 \)