Simplify expressions using the power rule of exponents.

Of course! Let's solve this worksheet step by step using the Power Rule of exponents.

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## 🔑 Power Rule Review:

The Power Rule states:
> When you raise a power to another power, you multiply the exponents.

In formula form:
> (a^m)^n = a^(m·n)

Also, when you have a product inside parentheses raised to a power, you apply the exponent to each factor:
> (a^m · b^n)^p = a^(m·p) · b^(n·p)

And remember:
> Any non-zero number (or variable) raised to the 0th power is 1:
> a^0 = 1, as long as a ≠ 0

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Now let’s solve each problem:

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✔ Left Column:

#### 1. `(x²)²`
→ Multiply exponents: 2 × 2 = 4
Answer: x⁴

#### 2. `(x²)³`
→ 2 × 3 = 6
Answer: x⁶

#### 3. `(x²)⁶`
→ 2 × 6 = 12
Answer: x¹²

#### 4. `(y³)²`
→ 3 × 2 = 6
Answer: y⁶

#### 5. `(x⁴)⁰`
→ Anything (non-zero) to the 0 power is 1
Answer: 1

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✔ Right Column:

#### 1. `(x²y²)³`
→ Apply exponent 3 to both x² and y²:
x^(2×3) · y^(2×3) = x⁶ · y⁶
Answer: x⁶y⁶

#### 2. `(x²y⁴)⁵`
→ x^(2×5) · y^(4×5) = x¹⁰ · y²⁰
Answer: x¹⁰y²⁰

#### 3. `(xy⁴)³`
→ x^(1×3) · y^(4×3) = x³ · y¹²
Answer: x³y¹²

#### 4. `(xy⁶)⁴`
→ x^(1×4) · y^(6×4) = x⁴ · y²⁴
Answer: x⁴y²⁴

#### 5. `(x⁰)⁵`
→ First, x⁰ = 1 (as long as x ≠ 0), then 1⁵ = 1
Answer: 1

*(Alternatively, using power rule: (x⁰)⁵ = x^(0×5) = x⁰ = 1)*

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## 🧾 Final Answers:

Left Column:

1. `(x²)² = x⁴`
2. `(x²)³ = x⁶`
3. `(x²)⁶ = x¹²`
4. `(y³)² = y⁶`
5. `(x⁴)⁰ = 1`

Right Column:

1. `(x²y²)³ = x⁶y⁶`
2. `(x²y⁴)⁵ = x¹⁰y²⁰`
3. `(xy⁴)³ = x³y¹²`
4. `(xy⁶)⁴ = x⁴y²⁴`
5. `(x⁰)⁵ = 1`

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✔ All problems solved using the Power Rule and basic exponent rules!

Let me know if you’d like to practice more or need a visual explanation!