Exponents Worksheet: Evaluate to a Single Number

Sure! Let’s solve each problem on the Exponents Worksheet step by step and explain the rules used.

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🔢 Exponent Rules Used:

- Positive exponent: \( a^n = a \times a \times \dots \times a \) (n times)
- Zero exponent: \( a^0 = 1 \) (for any \( a \neq 0 \))
- Negative exponent: \( a^{-n} = \frac{1}{a^n} \)
- Negative base with even/odd exponent:
- Even exponent → result is positive
- Odd exponent → result is negative

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## ✔ Solutions:

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1. \( 3^3 = \)

→ Multiply 3 × 3 × 3
= 27

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2. \( 5^4 = \)

→ 5 × 5 × 5 × 5
= 25 × 25 = 625

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3. \( 7^3 = \)

→ 7 × 7 × 7 = 49 × 7
= 343

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4. \( (-4)^3 = \)

→ Negative base, odd exponent → result is negative
= (-4) × (-4) × (-4) = 16 × (-4)
= -64

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5. \( (-8)^4 = \)

→ Negative base, even exponent → result is positive
= (-8) × (-8) × (-8) × (-8) = 64 × 64
= 4096

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6. \( (-6)^3 = \)

→ Negative base, odd exponent → result is negative
= (-6) × (-6) × (-6) = 36 × (-6)
= -216

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7. \( 8^0 = \)

→ Any non-zero number to the power of 0 is 1
= 1

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8. \( 15^{-3} = \)

→ Negative exponent: flip to reciprocal
= \( \frac{1}{15^3} = \frac{1}{15 \times 15 \times 15} = \frac{1}{3375} \)
= \(\frac{1}{3375}\)

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9. \( 12^{-4} = \)

→ \( \frac{1}{12^4} = \frac{1}{12 \times 12 \times 12 \times 12} = \frac{1}{20736} \)
= \(\frac{1}{20736}\)

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10. \( (-5)^0 = \)

→ Any non-zero number to the power of 0 is 1
= 1

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11. \( 0.01^{-1} = \)

→ First, note that \( 0.01 = \frac{1}{100} \), so
\( \left( \frac{1}{100} \right)^{-1} = 100 \)
OR: \( a^{-1} = \frac{1}{a} \), so \( \frac{1}{0.01} = 100 \)
= 100

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12. \( (-9)^{-1} = \)

→ \( \frac{1}{-9} = -\frac{1}{9} \)
= \(-\frac{1}{9}\)

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## 📝 Final Answers:

1. 27
2. 625
3. 343
4. -64
5. 4096
6. -216
7. 1
8. \(\frac{1}{3375}\)
9. \(\frac{1}{20736}\)
10. 1
11. 100
12. \(-\frac{1}{9}\)

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✔ All problems evaluated to a single number as requested!

Let me know if you want a printable version or further explanations!