Exponents Worksheet for practicing exponent rules and simplification.

Since I can’t see the image you uploaded, I’ll solve the Exponents Worksheet based on the text you provided — which appears to be a standard middle/high school exponent rules practice sheet.

I’ll go problem by problem, evaluate or simplify using exponent rules.

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🔢 Exponent Rules Recap (for reference):

- Product Rule: \( a^m \cdot a^n = a^{m+n} \)
- Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \)
- Power of a Power: \( (a^m)^n = a^{m \cdot n} \)
- Power of a Product: \( (ab)^n = a^n b^n \)
- Zero Exponent: \( a^0 = 1 \) (if \( a \ne 0 \))
- Negative Exponent: \( a^{-n} = \frac{1}{a^n} \)
- Distributive Property for Multiplication: \( a(b + c) = ab + ac \)

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## ✔ Problem-by-Problem Solutions:

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1. \( 3^2 + 5^2 \)

→ Evaluate each term:
\( 3^2 = 9 \), \( 5^2 = 25 \)
→ Add: \( 9 + 25 = 34 \)
✔ Answer: 34

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2. \( 4^4 + 6^3 \)

→ \( 4^4 = 4 \cdot 4 \cdot 4 \cdot 4 = 256 \)
→ \( 6^3 = 6 \cdot 6 \cdot 6 = 216 \)
→ Add: \( 256 + 216 = 472 \)
✔ Answer: 472

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3. \( 7^1 + 5^0 \)

→ \( 7^1 = 7 \)
→ \( 5^0 = 1 \) (any non-zero number to the power 0 is 1)
→ Add: \( 7 + 1 = 8 \)
✔ Answer: 8

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

⚠️ Important: Order matters!
- \( -4^3 \) means \( -(4^3) = -64 \) (not \( (-4)^3 \))
- \( 3^4 = 81 \)
→ So: \( -64 + 81 = 17 \)
✔ Answer: 17

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5. \( 8^4 \times 3^{-2} \)

→ \( 8^4 = (2^3)^4 = 2^{12} = 4096 \)
→ \( 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \)
→ Multiply: \( 4096 \times \frac{1}{9} = \frac{4096}{9} \)
✔ Answer: \( \frac{4096}{9} \) (or ≈ 455.11 as decimal)

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6. \( 6^{-3} \times 4^{-5} \)

→ \( 6^{-3} = \frac{1}{6^3} = \frac{1}{216} \)
→ \( 4^{-5} = \frac{1}{4^5} = \frac{1}{1024} \)
→ Multiply: \( \frac{1}{216} \times \frac{1}{1024} = \frac{1}{221184} \)
✔ Answer: \( \frac{1}{221184} \)

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7. \( (8x)^0 (12x)^3 \)

→ Anything (≠0) to the power 0 is 1 → \( (8x)^0 = 1 \)
→ So: \( 1 \cdot (12x)^3 = (12x)^3 = 12^3 \cdot x^3 = 1728x^3 \)
✔ Answer: \( 1728x^3 \)

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8. \( (12x)^{-1} (13x)^2 \)

→ \( (12x)^{-1} = \frac{1}{12x} \)
→ \( (13x)^2 = 169x^2 \)
→ Multiply: \( \frac{1}{12x} \cdot 169x^2 = \frac{169x^2}{12x} = \frac{169x}{12} \)
✔ Answer: \( \frac{169x}{12} \)

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9. \( (21x)^{-1} (14x)^2 \)

→ \( (21x)^{-1} = \frac{1}{21x} \)
→ \( (14x)^2 = 196x^2 \)
→ Multiply: \( \frac{196x^2}{21x} = \frac{196}{21} x = \frac{28}{3}x \) (simplify 196 ÷ 7 = 28, 21 ÷ 7 = 3)
✔ Answer: \( \frac{28}{3}x \)

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10. \( (34x)^0 (19x)^3 \)

→ \( (34x)^0 = 1 \)
→ \( (19x)^3 = 6859x^3 \)
✔ Answer: \( 6859x^3 \)

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11. \( f(x) = (-2x)^2 + (4x) \)

→ Simplify:
\( (-2x)^2 = (-2)^2 \cdot x^2 = 4x^2 \)
→ So: \( f(x) = 4x^2 + 4x \)
✔ Answer: \( 4x^2 + 4x \)

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12. \( f(x) = (22x)^0 + (11x)^2 \)

→ \( (22x)^0 = 1 \)
→ \( (11x)^2 = 121x^2 \)
→ So: \( f(x) = 1 + 121x^2 \)
✔ Answer: \( 121x^2 + 1 \)

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## 📝 Final Answers Summary:

| Problem | Answer |
|---------|---------------------|
| 1 | 34 |
| 2 | 472 |
| 3 | 8 |
| 4 | 17 |
| 5 | \( \frac{4096}{9} \) |
| 6 | \( \frac{1}{221184} \) |
| 7 | \( 1728x^3 \) |
| 8 | \( \frac{169x}{12} \) |
| 9 | \( \frac{28}{3}x \) |
| 10 | \( 6859x^3 \) |
| 11 | \( 4x^2 + 4x \) |
| 12 | \( 121x^2 + 1 \) |

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Let me know if you’d like these explained in more detail, or if you want to check your own answers against this key! 😊