Laws of Exponents worksheet - Free Printable

Let’s go through each problem one by one, simplifying the expressions so that all exponents are positive. Then we’ll match the solution to the coloring instructions.

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Problem 1: 9c⁵ · 5c⁻⁵

- Multiply coefficients: 9 × 5 = 45
- Add exponents of c: c⁵ · c⁻⁵ = c^(5 + (-5)) = c⁰ = 1
- So answer is 45 × 1 = 45
- Coloring instruction: Entire solution → purple

✔ Final Answer for #1: 45

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Problem 2: 2k⁶g² · 3kg³

- Coefficients: 2 × 3 = 6
- k terms: k⁶ · k¹ = k^(6+1) = k⁷
- g terms: g² · g³ = g^(2+3) = g⁵
- Solution: 6k⁷g⁵
- Coloring: Coefficient (yellow), exponent of g (dark blue)

✔ Final Answer for #2: 6k⁷g⁵

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Problem 3: 7 · 7⁻⁵

- Same base? Yes → add exponents: 7¹ · 7⁻⁵ = 7^(1 + (-5)) = 7⁻⁴
- But we need positive exponents → move to denominator: 1 / 7⁴
- Solution: 1/7⁴
- Coloring: Exponent in denominator → red

✔ Final Answer for #3: 1/7⁴

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Problem 4: (5h²)/(8h⁴)

- Coefficients stay: 5/8
- h terms: h² / h⁴ = h^(2-4) = h⁻² → move to denominator → becomes h² in denominator
- So: 5/(8h²)
- Coloring: Coefficient in denominator (pink) → that’s 8; exponent of h (light blue) → that’s 2

✔ Final Answer for #4: 5/(8h²)

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Problem 5: (5s⁻²n⁻⁵)/(7sn⁻⁶)

- Break it down:
- Coefficients: 5/7
- s terms: s⁻² / s¹ = s^(-2-1) = s⁻³ → move to denominator → s³
- n terms: n⁻⁵ / n⁻⁶ = n^(-5 - (-6)) = n¹ → stays in numerator
- So: (5n)/(7s³)
- Coloring: Variable in numerator → that’s ‘n’ → purple

✔ Final Answer for #5: (5n)/(7s³)

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Problem 6: (5w²)/(9w⁶y⁴)

- Coefficients: 5/9
- w terms: w² / w⁶ = w^(2-6) = w⁻⁴ → move to denominator → w⁴
- y term: already in denominator → y⁴
- So: 5/(9w⁴y⁴)
- Coloring: Sum of coefficient in denominator (9) and exponent of w (4) → 9 + 4 = 13 → dark blue

✔ Final Answer for #6: 5/(9w⁴y⁴)

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Problem 7: 6⁻³ / 6

- Denominator is 6¹ → so 6⁻³ / 6¹ = 6^(-3 - 1) = 6⁻⁴
- Positive exponent → 1 / 6⁴
- Coloring: Denominator simplified → that’s 6⁴ → red

✔ Final Answer for #7: 1/6⁴

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Problem 8: (1/n)⁶ · (1/n)³ · (1/n)⁵

- All same base → multiply by adding exponents: (1/n)^(6+3+5) = (1/n)^14
- Which is 1/n¹⁴
- Coloring: Power of the base → that’s 14 → light blue

✔ Final Answer for #8: 1/n¹⁴

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Problem 9: bs · 8b³s⁵

- Coefficients: 1 × 8 = 8
- b terms: b¹ · b³ = b⁴
- s terms: s¹ · s⁵ = s⁶
- Solution: 8b⁴s⁶
- Coloring: Sum of exponents of b and s → 4 + 6 = 10 → yellow

✔ Final Answer for #9: 8b⁴s⁶

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Problem 10: (6d⁻⁵h²)/(5d³h⁻³)

- Coefficients: 6/5
- d terms: d⁻⁵ / d³ = d^(-5-3) = d⁻⁸ → move to denominator → d⁸
- h terms: h² / h⁻³ = h^(2 - (-3)) = h⁵ → stays in numerator
- So: (6h⁵)/(5d⁸)
- Coloring: Variable in denominator → that’s ‘d’ → pink

✔ Final Answer for #10: (6h⁵)/(5d⁸)

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Problem 11: 2⁴ / 2⁶

- Same base → subtract exponents: 2^(4-6) = 2⁻² → move to denominator → 1/2²
- Coloring: Numerator → that’s 1 → purple

Wait — let’s double-check. The expression is 2⁴ / 2⁶ = 1 / 2². The numerator of the simplified fraction is 1.

But maybe they mean the original numerator? Let’s read carefully: “Numerator (purple)” — likely refers to the numerator of the final simplified expression.

In 1/2², numerator is 1 → color 1 purple.

✔ Final Answer for #11: 1/2²

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Problem 12: s² · s⁻⁵ · s⁻⁴

- Add exponents: 2 + (-5) + (-4) = -7
- So s⁻⁷ → write as 1/s⁷
- Coloring: Exponent of the denominator → that’s 7 → pink

✔ Final Answer for #12: 1/s⁷

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Now, compiling all answers clearly:

Final Answer:
1. 45
2. 6k⁷g⁵
3. 1/7⁴
4. 5/(8h²)
5. (5n)/(7s³)
6. 5/(9w⁴y⁴)
7. 1/6⁴
8. 1/n¹⁴
9. 8b⁴s⁶
10. (6h⁵)/(5d⁸)
11. 1/2²
12. 1/s⁷