Math worksheet for simplifying expressions by dividing powers with coefficients and the same base.

You're working on a worksheet titled “Dividing powers with coefficients and the same base”. The goal is to simplify expressions by dividing monomials — that means dividing both the coefficients (the numbers) and the variables with exponents using exponent rules.

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## 📌 Key Rules to Remember:

1. Dividing coefficients: Just divide the numbers normally.
2. Dividing variables with same base: Subtract the exponents → `x^m ÷ x^n = x^(m-n)`
3. Multiplying monomials (problems 10–15): Multiply coefficients, add exponents of same bases → `(ax^m)(bx^n) = ab·x^(m+n)`

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Let’s solve each problem step-by-step.

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✔ Problem 1: `18p⁸ ÷ 6p⁵`

- Coefficients: `18 ÷ 6 = 3`
- Variables: `p⁸ ÷ p⁵ = p^(8-5) = p³`
✔ Answer: `3p³`

*(Already shown in your image — correct!)*

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✔ Problem 2: `12a⁵ ÷ 3a³`

- Coefficients: `12 ÷ 3 = 4`
- Variables: `a⁵ ÷ a³ = a^(5-3) = a²`
✔ Answer: `4a²`

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✔ Problem 3: `28k⁷ ÷ 4k²`

- Coefficients: `28 ÷ 4 = 7`
- Variables: `k⁷ ÷ k² = k^(7-2) = k⁵`
✔ Answer: `7k⁵`

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✔ Problem 4: `10m⁹ ÷ 5m`

- Coefficients: `10 ÷ 5 = 2`
- Variables: `m⁹ ÷ m¹ = m^(9-1) = m⁸`
✔ Answer: `2m⁸`

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✔ Problem 5: `20c⁶ ÷ 2c³`

- Coefficients: `20 ÷ 2 = 10`
- Variables: `c⁶ ÷ c³ = c^(6-3) = c³`
✔ Answer: `10c³`

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✔ Problem 6: `90f⁸ ÷ 10f⁶`

- Coefficients: `90 ÷ 10 = 9`
- Variables: `f⁸ ÷ f⁶ = f^(8-6) = f²`
✔ Answer: `9f²`

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✔ Problem 7: `36w⁶ ÷ 9w²`

- Coefficients: `36 ÷ 9 = 4`
- Variables: `w⁶ ÷ w² = w^(6-2) = w⁴`
✔ Answer: `4w⁴`

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✔ Problem 8: `54y¹¹ / 9y⁷`

- Coefficients: `54 ÷ 9 = 6`
- Variables: `y¹¹ ÷ y⁷ = y^(11-7) = y⁴`
✔ Answer: `6y⁴`

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✔ Problem 9: `12b³ / 3b`

- Coefficients: `12 ÷ 3 = 4`
- Variables: `b³ ÷ b¹ = b^(3-1) = b²`
✔ Answer: `4b²`

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## 🔁 Now for Multiplication Problems (10–15):

These are multiplying monomials, not dividing — but still use coefficient multiplication and exponent addition.

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✔ Problem 10: `(4x⁵)(-3x²)`

- Coefficients: `4 × (-3) = -12`
- Variables: `x⁵ × x² = x^(5+2) = x⁷`
✔ Answer: `-12x⁷`

*(Your image shows this correctly — good job!)*

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✔ Problem 11: `(5x⁵y)(3x²y²)`

- Coefficients: `5 × 3 = 15`
- Variables:
- `x⁵ × x² = x⁷`
- `y¹ × y² = y³`
✔ Answer: `15x⁷y³`

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✔ Problem 12: `(-5y³)(2y⁴)`

- Coefficients: `-5 × 2 = -10`
- Variables: `y³ × y⁴ = y^(3+4) = y⁷`
✔ Answer: `-10y⁷`

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✔ Problem 13: `(9a⁴b)(2ab³)`

- Coefficients: `9 × 2 = 18`
- Variables:
- `a⁴ × a¹ = a⁵`
- `b¹ × b³ = b⁴`
✔ Answer: `18a⁵b⁴`

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✔ Problem 14: `(-5c⁵d³)(8c²d²)`

- Coefficients: `-5 × 8 = -40`
- Variables:
- `c⁵ × c² = c⁷`
- `d³ × d² = d⁵`
✔ Answer: `-40c⁷d⁵`

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✔ Problem 15: `(9x⁶y²)(3y)`

- Coefficients: `9 × 3 = 27`
- Variables:
- `x⁶` (no other x → stays `x⁶`)
- `y² × y¹ = y³`
✔ Answer: `27x⁶y³`

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## 🧾 Final Answers Summary:

| Problem | Answer |
|---------|---------------|
| 1 | `3p³` |
| 2 | `4a²` |
| 3 | `7k⁵` |
| 4 | `2m⁸` |
| 5 | `10c³` |
| 6 | `9f²` |
| 7 | `4w⁴` |
| 8 | `6y⁴` |
| 9 | `4b²` |
| 10 | `-12x⁷` |
| 11 | `15x⁷y³` |
| 12 | `-10y⁷` |
| 13 | `18a⁵b⁴` |
| 14 | `-40c⁷d⁵` |
| 15 | `27x⁶y³` |

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✔ All problems solved using basic exponent rules and arithmetic. Great practice for simplifying algebraic expressions!

Let me know if you want a printable version or more practice problems!