Let’s solve each problem step by step using the rule for multiplying exponents with the same base:
>
When you multiply powers with the same base, add the exponents.
> That is: \( a^m \times a^n = a^{m+n} \)
We’ll apply this to each problem.
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1. \( 8^{-14} \times 8^{-10} \)
Same base (8), so add exponents:
\( -14 + (-10) = -24 \)
→ \( 8^{-24} \)
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2. \( 6^3 \times 6^{-1} \times 6^{-3} \)
Add all exponents:
\( 3 + (-1) + (-3) = 3 - 1 - 3 = -1 \)
→ \( 6^{-1} \)
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3. \( 5^{-1} \times 5^{-3} \times 5^{-4} \)
Add exponents:
\( -1 + (-3) + (-4) = -8 \)
→ \( 5^{-8} \)
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4. \( (-3)^3 \times (-3)^{-2} \)
Same base (-3), add exponents:
\( 3 + (-2) = 1 \)
→ \( (-3)^1 = -3 \)
*(Note: The base is negative, but since we’re just adding exponents and not changing the base, it stays as -3 raised to power 1.)*
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5. \( 4^2 \times 4^3 \times 4^{-5} \)
Add exponents:
\( 2 + 3 + (-5) = 0 \)
→ \( 4^0 = 1 \) *(Any nonzero number to the 0 power is 1)*
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6. \( 5^3 \times 5^6 \)
Add exponents:
\( 3 + 6 = 9 \)
→ \( 5^9 \)
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7. \( 8^3 \times 8^2 \times 8^{-4} \)
Add exponents:
\( 3 + 2 + (-4) = 1 \)
→ \( 8^1 = 8 \)
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8. \( 9^2 \times 9^{-2} \times 9^4 \)
Add exponents:
\( 2 + (-2) + 4 = 4 \)
→ \( 9^4 \)
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9. \( 7^3 \times 7^{-2} \)
Add exponents:
\( 3 + (-2) = 1 \)
→ \( 7^1 = 7 \)
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10. \( 2^3 \times 2^{-4} \times 2^3 \times 2^6 \)
Add all exponents:
\( 3 + (-4) + 3 + 6 = (3 + 3 + 6) - 4 = 12 - 4 = 8 \)
→ \( 2^8 \)
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Final Answer:
1. \( 8^{-24} \)
2. \( 6^{-1} \)
3. \( 5^{-8} \)
4. \( -3 \)
5. \( 1 \)
6. \( 5^9 \)
7. \( 8 \)
8. \( 9^4 \)
9. \( 7 \)
10. \( 2^8 \)
Parent Tip: Review the logic above to help your child master the concept of multiplying with exponents worksheet.
