Grade 9 Mathematics worksheet on Laws of Exponents with simplification exercises.

Let's go through each problem on the worksheet and verify or explain how to simplify using the Laws of Exponents. The goal is to check whether each simplification is correct and explain the reasoning.

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🔷 Laws of Exponents Recap:

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

Now let’s analyze each problem:

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1. $ 4x^3 \cdot 2x^3 = 8x^6 $

✔ Correct

- Multiply coefficients: $ 4 \cdot 2 = 8 $
- Add exponents of same base $ x $: $ x^3 \cdot x^3 = x^{3+3} = x^6 $
- So: $ 8x^6 $

✔️ Correct answer

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2. $ x^5 \cdot x^3 = x^8 $

✔ Correct

- Use product rule: $ x^5 \cdot x^3 = x^{5+3} = x^8 $

✔️ Correct answer

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3. $ 2x^3 \cdot 2x^2 = 4x^5 $

✔ Correct

- Coefficients: $ 2 \cdot 2 = 4 $
- Variables: $ x^3 \cdot x^2 = x^{3+2} = x^5 $
- Result: $ 4x^5 $

✔️ Correct answer

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4. $ \frac{6^5}{6^3} = 36 $

✘ Incorrect

- Use quotient rule: $ \frac{6^5}{6^3} = 6^{5-3} = 6^2 = 36 $
- Wait — $ 6^2 = 36 $, so actually this is correct!

But let's double-check:
- $ 6^5 = 7776 $
- $ 6^3 = 216 $
- $ \frac{7776}{216} = 36 $

✔ So yes, it is correct.

✔️ Correct answer

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5. $ \frac{x^4}{x^7} = \frac{1}{x^3} $

✔ Correct

- Quotient rule: $ \frac{x^4}{x^7} = x^{4-7} = x^{-3} = \frac{1}{x^3} $

✔️ Correct answer

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6. $ 8^0 = 1 $

✔ Correct

- Any non-zero number raised to power 0 is 1.

✔️ Correct answer

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7. $ \frac{6x^7}{2x^4} = 3x^3 $

✔ Correct

- Simplify coefficients: $ \frac{6}{2} = 3 $
- Simplify variables: $ \frac{x^7}{x^4} = x^{7-4} = x^3 $
- Result: $ 3x^3 $

✔️ Correct answer

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8. $ \frac{x^2 y^5}{x y^4} = xy $

✔ Correct

- Simplify $ x $: $ \frac{x^2}{x} = x^{2-1} = x $
- Simplify $ y $: $ \frac{y^5}{y^4} = y^{5-4} = y $
- So: $ x \cdot y = xy $

✔️ Correct answer

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✔ Final Answer Summary:

All eight problems are correctly simplified!
Each one applies the appropriate law of exponents properly.

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📌 Summary of Key Concepts Used:

| Problem | Law Used |
|--------|----------|
| 1, 3 | Product of Powers & Coefficient Multiplication |
| 2 | Product of Powers |
| 4 | Quotient of Powers |
| 5 | Quotient of Powers + Negative Exponent |
| 6 | Zero Exponent Rule |
| 7 | Quotient of Powers + Coefficient Division |
| 8 | Quotient of Powers for multiple variables |

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🎉 All answers on the worksheet are correct!
This is a well-designed exercise reinforcing exponent rules in Grade 9 math.