Free Guide: Dividing Exponents Explained — Mashup Math - Free Printable

You’ve uploaded a worksheet titled “Practice: Dividing Exponents”, and it contains 20 problems asking you to simplify expressions involving multiplication and division of powers with the same base.

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## 🔑 Key Rules to Remember:

When simplifying expressions with exponents, especially when multiplying or dividing powers with the same base, use these rules:

1. Multiplying Powers (Same Base):

> \( a^m \cdot a^n = a^{m+n} \)

2. Dividing Powers (Same Base):

> \( \frac{a^m}{a^n} = a^{m-n} \) (if \( m > n \))

> \( \frac{a^m}{a^n} = \frac{1}{a^{n-m}} \) (if \( n > m \))

> \( \frac{a^m}{a^m} = 1 \)

3. Power of 1:

> \( a^1 = a \)

4. Power of 0:

> \( a^0 = 1 \) (for \( a \ne 0 \))

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Let’s solve each problem step-by-step.

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✔ Problem 1:

\( \frac{3^5 \cdot 3^4}{3^4} \)

→ Multiply numerator: \( 3^{5+4} = 3^9 \)
→ Divide: \( \frac{3^9}{3^4} = 3^{9-4} = 3^5 \)

Answer: \( \boxed{3^5} \)

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✔ Problem 2:

\( \frac{5^4 \cdot 5^3}{5^3} \)

→ Numerator: \( 5^{4+3} = 5^7 \)
→ Divide: \( \frac{5^7}{5^3} = 5^{7-3} = 5^4 \)

Answer: \( \boxed{5^4} \)

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✔ Problem 3:

\( \frac{3^1 \cdot 3^4}{3^3} \)

→ Numerator: \( 3^{1+4} = 3^5 \)
→ Divide: \( \frac{3^5}{3^3} = 3^{5-3} = 3^2 \)

Answer: \( \boxed{3^2} \)

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✔ Problem 4:

\( \frac{5}{5^3 \cdot 5^2} \)

→ Denominator: \( 5^{3+2} = 5^5 \)
→ So: \( \frac{5^1}{5^5} = 5^{1-5} = 5^{-4} = \frac{1}{5^4} \)

Answer: \( \boxed{\frac{1}{5^4}} \)

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✔ Problem 5:

\( \frac{6 \cdot 6^4}{6} \)

→ Numerator: \( 6^1 \cdot 6^4 = 6^5 \)
→ Divide: \( \frac{6^5}{6^1} = 6^{5-1} = 6^4 \)

Answer: \( \boxed{6^4} \)

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✔ Problem 6:

\( \frac{2^2 \cdot 2^4}{2} \)

→ Numerator: \( 2^{2+4} = 2^6 \)
→ Divide: \( \frac{2^6}{2^1} = 2^{6-1} = 2^5 \)

Answer: \( \boxed{2^5} \)

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✔ Problem 7:

\( \frac{7^2 \cdot 7^4}{7} \)

→ Numerator: \( 7^{2+4} = 7^6 \)
→ Divide: \( \frac{7^6}{7^1} = 7^{6-1} = 7^5 \)

Answer: \( \boxed{7^5} \)

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✔ Problem 8:

\( \frac{64}{8^2} \)

→ Note: 64 = \( 8^2 \), since \( 8 \cdot 8 = 64 \)
→ So: \( \frac{8^2}{8^2} = 8^{2-2} = 8^0 = 1 \)

Answer: \( \boxed{1} \)

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✔ Problem 9:

\( \frac{8 \cdot 8^3}{8^4} \)

→ Numerator: \( 8^1 \cdot 8^3 = 8^4 \)
→ Divide: \( \frac{8^4}{8^4} = 8^{4-4} = 8^0 = 1 \)

Answer: \( \boxed{1} \)

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✔ Problem 10:

\( \frac{4}{4^3 \cdot 4^2} \)

→ Denominator: \( 4^{3+2} = 4^5 \)
→ So: \( \frac{4^1}{4^5} = 4^{1-5} = 4^{-4} = \frac{1}{4^4} \)

Answer: \( \boxed{\frac{1}{4^4}} \)

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✔ Problem 11:

\( \frac{7^2 \cdot 7^3}{7^3} \)

→ Numerator: \( 7^{2+3} = 7^5 \)
→ Divide: \( \frac{7^5}{7^3} = 7^{5-3} = 7^2 \)

Answer: \( \boxed{7^2} \)

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✔ Problem 12:

\( \frac{8^3 \cdot 8^2}{8} \)

→ Numerator: \( 8^{3+2} = 8^5 \)
→ Divide: \( \frac{8^5}{8^1} = 8^{5-1} = 8^4 \)

Answer: \( \boxed{8^4} \)

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✔ Problem 13:

\( \frac{3^3}{3 \cdot 3^4} \)

→ Denominator: \( 3^1 \cdot 3^4 = 3^5 \)
→ So: \( \frac{3^3}{3^5} = 3^{3-5} = 3^{-2} = \frac{1}{3^2} \)

Answer: \( \boxed{\frac{1}{3^2}} \)

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✔ Problem 14:

\( \frac{5^2 \cdot 5^4 \cdot 5^2}{5^5} \)

→ Numerator: \( 5^{2+4+2} = 5^8 \)
→ Divide: \( \frac{5^8}{5^5} = 5^{8-5} = 5^3 \)

Answer: \( \boxed{5^3} \)

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✔ Problem 15:

\( \frac{2 \cdot 2^3}{2^2} \)

→ Numerator: \( 2^1 \cdot 2^3 = 2^4 \)
→ Divide: \( \frac{2^4}{2^2} = 2^{4-2} = 2^2 \)

Answer: \( \boxed{2^2} \)

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✔ Problem 16:

\( \frac{3^3}{3^2 \cdot 3^1} \)

→ Denominator: \( 3^{2+1} = 3^3 \)
→ So: \( \frac{3^3}{3^3} = 3^{3-3} = 3^0 = 1 \)

Answer: \( \boxed{1} \)

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✔ Problem 17:

\( \frac{8^2 \cdot 8^2}{8^4} \)

→ Numerator: \( 8^{2+2} = 8^4 \)
→ Divide: \( \frac{8^4}{8^4} = 8^{4-4} = 8^0 = 1 \)

Answer: \( \boxed{1} \)

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✔ Problem 18:

\( \frac{6^3 \cdot 6^4}{6^3} \)

→ Numerator: \( 6^{3+4} = 6^7 \)
→ Divide: \( \frac{6^7}{6^3} = 6^{7-3} = 6^4 \)

Answer: \( \boxed{6^4} \)

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✔ Problem 19:

\( \frac{8^5 \cdot 8^4}{8^4} \)

→ Numerator: \( 8^{5+4} = 8^9 \)
→ Divide: \( \frac{8^9}{8^4} = 8^{9-4} = 8^5 \)

Answer: \( \boxed{8^5} \)

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✔ Problem 20:

\( \frac{2 \cdot 2^4}{2^4} \)

→ Numerator: \( 2^1 \cdot 2^4 = 2^5 \)
→ Divide: \( \frac{2^5}{2^4} = 2^{5-4} = 2^1 = 2 \)

Answer: \( \boxed{2} \)

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## 🎯 Final Answer Key:

1. \( 3^5 \)
2. \( 5^4 \)
3. \( 3^2 \)
4. \( \frac{1}{5^4} \)
5. \( 6^4 \)
6. \( 2^5 \)
7. \( 7^5 \)
8. \( 1 \)
9. \( 1 \)
10. \( \frac{1}{4^4} \)
11. \( 7^2 \)
12. \( 8^4 \)
13. \( \frac{1}{3^2} \)
14. \( 5^3 \)
15. \( 2^2 \)
16. \( 1 \)
17. \( 1 \)
18. \( 6^4 \)
19. \( 8^5 \)
20. \( 2 \)

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✔ All problems solved using exponent rules! Let me know if you’d like a printable version or step-by-step explanations for any specific problem.