Powers of Products Worksheets - 15 Worksheets Library - Free Printable

You're working on simplifying expressions using the Power of a Product Rule.

🔑 Key Concept: Power of a Product Rule

When you raise a product (multiplication) to a power, you apply that power to each factor inside the parentheses:

> (a · b · c)ⁿ = aⁿ · bⁿ · cⁿ

Also, remember the Product of Powers Rule for same bases:
> xᵃ · xᵇ = xᵃ⁺ᵇ

And the Power of a Power Rule:
> (xᵃ)ᵇ = xᵃᵇ

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Let’s solve each problem step by step.

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## ✔ Problem 1: (x³ · x⁴ · 2x³)³ → Given answer: (8x³⁰)

Step 1: Simplify inside the parentheses first.

Group like terms:
- Coefficients: only one coefficient → 2
- Variables: x³ · x⁴ · x³ = x^(3+4+3) = x¹⁰

So inside: 2x¹⁰

Step 2: Apply the outer exponent (³) to both the coefficient and the variable.

→ (2x¹⁰)³ = 2³ · (x¹⁰)³ = 8 · x³⁰ = 8x³⁰ ✔️

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## ✔ Problem 2: (3b³ · b⁴ · b⁷)⁴

Step 1: Simplify inside.

Coefficients: 3
Variables: b³ · b⁴ · b⁷ = b^(3+4+7) = b¹⁴

Inside: 3b¹⁴

Step 2: Raise to the 4th power.

→ (3b¹⁴)⁴ = 3⁴ · (b¹⁴)⁴ = 81 · b⁵⁶ = 81b⁵⁶

✔ Answer: 81b⁵⁶

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## ✔ Problem 3: (a³ · 5a⁴ · 2a³)⁵

Step 1: Inside parentheses:

Coefficients: 5 · 2 = 10
Variables: a³ · a⁴ · a³ = a^(3+4+3) = a¹⁰

Inside: 10a¹⁰

Step 2: Raise to 5th power.

→ (10a¹⁰)⁵ = 10⁵ · (a¹⁰)⁵ = 100,000 · a⁵⁰ = 100000a⁵⁰

✔ Answer: 100000a⁵⁰

*(Note: You can write it as 10⁵a⁵⁰ if preferred)*

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## ✔ Problem 4: (3c² · 2c³ · c²)²

Step 1: Inside:

Coefficients: 3 · 2 = 6
Variables: c² · c³ · c² = c^(2+3+2) = c⁷

Inside: 6c⁷

Step 2: Square it.

→ (6c⁷)² = 6² · (c⁷)² = 36 · c¹⁴ = 36c¹⁴

✔ Answer: 36c¹⁴

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## ✔ Problem 5: (2a² · 3a² · 2a³)³

Step 1: Inside:

Coefficients: 2 · 3 · 2 = 12
Variables: a² · a² · a³ = a^(2+2+3) = a⁷

Inside: 12a⁷

Step 2: Cube it.

→ (12a⁷)³ = 12³ · (a⁷)³ = 1728 · a²¹ = 1728a²¹

✔ Answer: 1728a²¹

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## ✔ Problem 6: (2z³ · 2z⁴ · z³)³

Step 1: Inside:

Coefficients: 2 · 2 = 4
Variables: z³ · z⁴ · z³ = z^(3+4+3) = z¹⁰

Inside: 4z¹⁰

Step 2: Cube it.

→ (4z¹⁰)³ = 4³ · (z¹⁰)³ = 64 · z³⁰ = 64z³⁰

✔ Answer: 64z³⁰

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## ✔ Problem 7: (3x²y³ · 5y⁴ · 2xy³)²

Step 1: Multiply inside — group coefficients and variables separately.

Coefficients: 3 · 5 · 2 = 30

Variables:
- x: x² · x = x³
- y: y³ · y⁴ · y³ = y^(3+4+3) = y¹⁰

Inside: 30x³y¹⁰

Step 2: Square it.

→ (30x³y¹⁰)² = 30² · (x³)² · (y¹⁰)² = 900 · x⁶ · y²⁰ = 900x⁶y²⁰

✔ Answer: 900x⁶y²⁰

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## ✔ Problem 8: (3x²yz³ · z⁴ · 2xy³)⁴

Step 1: Multiply inside.

Coefficients: 3 · 2 = 6

Variables:
- x: x² · x = x³
- y: y · y³ = y⁴
- z: z³ · z⁴ = z⁷

Inside: 6x³y⁴z⁷

Step 2: Raise to 4th power.

→ (6x³y⁴z⁷)⁴ = 6⁴ · (x³)⁴ · (y⁴)⁴ · (z⁷)⁴
= 1296 · x¹² · y¹⁶ · z²⁸ = 1296x¹²y¹⁶z²⁸

✔ Answer: 1296x¹²y¹⁶z²⁸

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## 📝 Final Answers:

| Problem | Answer |
|---------|----------------------------|
| 1 | 8x³⁰ |
| 2 | 81b⁵⁶ |
| 3 | 100000a⁵⁰ |
| 4 | 36c¹⁴ |
| 5 | 1728a²¹ |
| 6 | 64z³⁰ |
| 7 | 900x⁶y²⁰ |
| 8 | 1296x¹²y¹⁶z²⁸ |

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## 💡 Summary of Strategy:

1. Multiply coefficients inside the parentheses.
2. Add exponents of same variables (Product of Powers).
3. Apply the outer exponent to every factor (Power of a Product).
4. Use Power of a Power rule for variables: (x^a)^b = x^(a·b).

This method ensures you simplify correctly every time!

Let me know if you’d like to see any step broken down further!