Integer Exponents

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Concept Review

The Product Rule: When Powers Multiply Their Strength

Imagine you're building with identical LEGO blocks. If you have 2³ blocks in one pile and 2⁴ blocks in another pile, how many blocks do you have total when you combine them? The answer isn't 2⁷ — it's much bigger than that! But what if you're multiplying the piles instead of adding them?

When we multiply powers that have the same base, something elegant happens. Instead of doing all the multiplication by hand, we can use a shortcut called the Product Rule for Exponents.

The Foundation: What Exponents Really Mean

Before we jump into the rule, let's remember that exponents are just repeated multiplication:

• 3² means 3 × 3 = 9
• 3⁴ means 3 × 3 × 3 × 3 = 81
• 5³ means 5 × 5 × 5 = 125

Discovering the Pattern

Let's multiply 3² × 3⁴ step by step:

3² × 3⁴ = (3 × 3) × (3 × 3 × 3 × 3) = 3 × 3 × 3 × 3 × 3 × 3

Count the 3's: we have six of them! So 3² × 3⁴ = 3⁶

🔑 The "Aha" Moment

Notice something amazing: 2 + 4 = 6, and 3² × 3⁴ = 3⁶

When multiplying powers with the same base, you ADD the exponents!

This works because you're literally counting up all the repeated multiplications.

The Product Rule in Action

The rule is: aᵐ × aⁿ = aᵐ⁺ⁿ

Let's see it work with real numbers:

Example: 2³ × 2⁵

Method 1 (The Long Way): (2×2×2) × (2×2×2×2×2) = 8 × 32 = 256

Method 2 (Product Rule): 2³ × 2⁵ = 2³⁺⁵ = 2⁸ = 256

Same answer, way faster!

This rule works with any base:

• x⁷ × x³ = x¹⁰
• (-4)² × (-4)⁶ = (-4)⁸
• 10¹ × 10⁹ = 10¹⁰ (that's 10 billion!)

🔑 Key Takeaway

Just like those LEGO blocks, when you multiply powers with the same base, you're combining all the repeated multiplications. The product rule gives us a shortcut: add the exponents and keep the base the same. It's not magic — it's just smart counting!

Sample questions

1. Simplify 5⁴ × 5² using the product rule.

○ 5⁸

✓ 5⁶

○ 25⁶

○ 5²

Answer: 5⁶ — When multiplying with the same base, ADD the exponents: 4 + 2 = 6.

2. Which expression is equivalent to 2³ × 2⁵?

○ 2¹⁵

○ 4⁸

✓ 2⁸

○ 2²

Answer: 2⁸ — 3 + 5 = 8, so 2⁸.

3. A student writes x⁴ × x⁷ = x¹¹. Is this correct?

✓ Yes, because 4 + 7 = 11

○ No, because you multiply the exponents: 4 × 7 = 28

○ No, because you add the bases: x + x = 2x

○ Yes, because 4 × 7 = 28 but they wrote 11

Answer: Yes, because 4 + 7 = 11 — Product rule: add exponents when multiplying same base.

Skills in this topic

Apply the product rule for exponents with the same base
Apply the quotient rule for exponents with the same base
Apply the power of a power rule for exponents
Evaluate expressions with zero and negative integer exponents
Simplify complex numerical expressions using all properties of integer exponents

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