Problem: Solve the given problems involving multiplication of exponents.
The key rule for multiplying exponents with the same base is:
\[
a^m \times a^n = a^{m+n}
\]
This means when you multiply two powers with the same base, you add their exponents.
Let's solve each problem step by step:
---
Problem 1: \( 8^{-14} \times 8^{-10} \)
- Both terms have the same base (\(8\)).
- Add the exponents: \(-14 + (-10) = -24\).
- Result: \( 8^{-24} \).
Answer: \( 8^{-24} \)
---
Problem 2: \( 6^3 \times 6^{-1} \times 6^{-3} \)
- All terms have the same base (\(6\)).
- Add the exponents: \(3 + (-1) + (-3) = -1\).
- Result: \( 6^{-1} \).
Answer: \( 6^{-1} \)
---
Problem 3: \( 5^{-1} \times 5^3 \times 5^{-4} \)
- All terms have the same base (\(5\)).
- Add the exponents: \(-1 + 3 + (-4) = -2\).
- Result: \( 5^{-2} \).
Answer: \( 5^{-2} \)
---
Problem 4: \( (-3)^3 \times (-3)^{-2} \)
- Both terms have the same base (\(-3\)).
- Add the exponents: \(3 + (-2) = 1\).
- Result: \( (-3)^1 = -3 \).
Answer: \( -3 \)
---
Problem 5: \( 4^2 \times 4^3 \times 4^{-5} \)
- All terms have the same base (\(4\)).
- Add the exponents: \(2 + 3 + (-5) = 0\).
- Any number raised to the power of 0 is 1.
- Result: \( 4^0 = 1 \).
Answer: \( 1 \)
---
Problem 6: \( 5^3 \times 5^6 \)
- Both terms have the same base (\(5\)).
- Add the exponents: \(3 + 6 = 9\).
- Result: \( 5^9 \).
Answer: \( 5^9 \)
---
Problem 7: \( 8^3 \times 8^2 \times 8^{-4} \)
- All terms have the same base (\(8\)).
- Add the exponents: \(3 + 2 + (-4) = 1\).
- Result: \( 8^1 = 8 \).
Answer: \( 8 \)
---
Problem 8: \( 9^2 \times 9^2 \times 9^4 \)
- All terms have the same base (\(9\)).
- Add the exponents: \(2 + 2 + 4 = 8\).
- Result: \( 9^8 \).
Answer: \( 9^8 \)
---
Problem 9: \( 7^3 \times 7^2 \)
- Both terms have the same base (\(7\)).
- Add the exponents: \(3 + 2 = 5\).
- Result: \( 7^5 \).
Answer: \( 7^5 \)
---
Problem 10: \( 2^3 \times 2^4 \times 2^3 \times 2^{-6} \)
- All terms have the same base (\(2\)).
- Add the exponents: \(3 + 4 + 3 + (-6) = 4\).
- Result: \( 2^4 \).
Answer: \( 2^4 \)
---
Final Answers:
1. \( 8^{-24} \)
2. \( 6^{-1} \)
3. \( 5^{-2} \)
4. \( -3 \)
5. \( 1 \)
6. \( 5^9 \)
7. \( 8 \)
8. \( 9^8 \)
9. \( 7^5 \)
10. \( 2^4 \)
\boxed{
\begin{array}{ll}
1. & 8^{-24} \\
2. & 6^{-1} \\
3. & 5^{-2} \\
4. & -3 \\
5. & 1 \\
6. & 5^9 \\
7. & 8 \\
8. & 9^8 \\
9. & 7^5 \\
10. & 2^4 \\
\end{array}
}
Parent Tip: Review the logic above to help your child master the concept of multiplying exponents with same base worksheet.
