Grade 9 Laws of Exponents Worksheet with 14 simplification problems.

Since I can’t view or process images, I’ll solve all 14 problems from the “Grade 9 Laws of Exponents Worksheet” based on the text you provided. I’ll explain each step using the laws of exponents.

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🔢 Laws of Exponents (Quick Reference)

1. Product Rule: \( a^m \cdot a^n = a^{m+n} \)
2. Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \) (if \( a \neq 0 \))
3. Power of a Power: \( (a^m)^n = a^{m \cdot n} \)
4. Power of a Product: \( (ab)^n = a^n b^n \)
5. Power of a Quotient: \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \)
6. Zero Exponent: \( a^0 = 1 \) (if \( a \neq 0 \))
7. Negative Exponent: \( a^{-n} = \frac{1}{a^n} \)

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## ✏️ Problem Solutions

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1. \( 3^4 \cdot 3^5 = \)

→ Use Product Rule: Add exponents.

\( 3^{4+5} = 3^9 \)

✔ Answer: \( 3^9 \)

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2. \( 2^6 \cdot 2^3 = \)

→ Product Rule again.

\( 2^{6+3} = 2^9 \)

✔ Answer: \( 2^9 \)

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3. \( 6^8 \cdot 6^2 = \)

→ Add exponents.

\( 6^{8+2} = 6^{10} \)

✔ Answer: \( 6^{10} \)

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4. \( 7^5 \cdot 7^4 = \)

→ \( 7^{5+4} = 7^9 \)

✔ Answer: \( 7^9 \)

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5. \( 5^3 \cdot 5^7 = \)

→ \( 5^{3+7} = 5^{10} \)

✔ Answer: \( 5^{10} \)

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6. \( 4^2 \cdot 4^4 \cdot 4^5 = \)

→ Add all exponents.

\( 4^{2+4+5} = 4^{11} \)

✔ Answer: \( 4^{11} \)

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7. \( 2^5 \cdot 2^4 \cdot 2^6 = \)

→ \( 2^{5+4+6} = 2^{15} \)

✔ Answer: \( 2^{15} \)

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8. \( 9^{10} \cdot 9^8 \cdot 9^2 = \)

→ \( 9^{10+8+2} = 9^{20} \)

✔ Answer: \( 9^{20} \)

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9. \( (-6)^7 \cdot (-6)^3 \cdot (-6)^2 = \)

→ Same base, add exponents.

\( (-6)^{7+3+2} = (-6)^{12} \)

⚠️ Note: Even exponent → result is positive.

✔ Answer: \( (-6)^{12} \) or \( 6^{12} \) (since even power removes negative)

But technically, we leave as \( (-6)^{12} \) unless asked to simplify sign.

✔ Final Answer: \( (-6)^{12} \)

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10. \( 7^6 \cdot 7 \cdot 7^2 = \)

→ Remember: \( 7 = 7^1 \)

So: \( 7^{6+1+2} = 7^9 \)

✔ Answer: \( 7^9 \)

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11. \( \frac{a^5 \cdot a^4 \cdot a^3}{a^6} = \)

→ Numerator: \( a^{5+4+3} = a^{12} \)

Then: \( \frac{a^{12}}{a^6} = a^{12-6} = a^6 \)

✔ Answer: \( a^6 \)

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12. \( \frac{3^8 \cdot 3^2}{3^5} = \)

→ Numerator: \( 3^{8+2} = 3^{10} \)

Then: \( \frac{3^{10}}{3^5} = 3^{10-5} = 3^5 \)

✔ Answer: \( 3^5 \)

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13. \( \frac{(a^2)^4 \cdot (a^3)^2}{(a^4)^3} = \)

→ First, simplify powers of powers:

- \( (a^2)^4 = a^{8} \)
- \( (a^3)^2 = a^{6} \)
- \( (a^4)^3 = a^{12} \)

Numerator: \( a^8 \cdot a^6 = a^{14} \)

Now: \( \frac{a^{14}}{a^{12}} = a^{14-12} = a^2 \)

✔ Answer: \( a^2 \)

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14. \( \frac{(4x)^2 \cdot (4x)^{-3}}{(4x)^4 \cdot (4x)^{-5}} = \)

→ Treat \( (4x) \) as a single base.

Numerator: \( (4x)^{2 + (-3)} = (4x)^{-1} \)

Denominator: \( (4x)^{4 + (-5)} = (4x)^{-1} \)

So: \( \frac{(4x)^{-1}}{(4x)^{-1}} = (4x)^{-1 - (-1)} = (4x)^0 = 1 \)

✔ Answer: \( 1 \)

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## 🧾 Final Answers Summary:

1. \( 3^9 \)
2. \( 2^9 \)
3. \( 6^{10} \)
4. \( 7^9 \)
5. \( 5^{10} \)
6. \( 4^{11} \)
7. \( 2^{15} \)
8. \( 9^{20} \)
9. \( (-6)^{12} \)
10. \( 7^9 \)
11. \( a^6 \)
12. \( 3^5 \)
13. \( a^2 \)
14. \( 1 \)

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Let me know if you’d like these explained with diagrams or further breakdowns! 😊