Exponents and Powers: Repeated Multiplication Compressed

Exponents are a counting system for factors.

3⁴ doesn't mean "three to the fourth power" — it means "four 3s multiplied together." The exponent counts how many times the base appears. That's all it does.

Here's why this matters: once you see exponents as counting, every rule becomes obvious. Multiplying same-base exponents? You're combining factors, so add the counts. Dividing? You're canceling factors, so subtract. Zero exponent? Zero factors multiplied together is 1. Negative exponent? You owe factors, so they go in the denominator.

The rules aren't a list to memorize. They're consequences of counting.

3 × 3 × 3 × 3 = 3⁴ = 81

The Definition

aⁿ means "multiply a by itself n times."

2³ = 2 × 2 × 2 = 8 (three 2s) 5² = 5 × 5 = 25 (two 5s) 10⁴ = 10 × 10 × 10 × 10 = 10,000 (four 10s)

a is the base — what you're multiplying. n is the exponent (or power) — how many times.

Why Exponents Matter

Exponents compress big numbers:

10¹² is easier to write than 1,000,000,000,000

They also describe growth:

• 2¹⁰ = 1,024 (exponential growth — doubling 10 times)
• 10⁶ bacteria after 6 doublings

And they appear throughout math and science:

• Area of a square: s²
• Volume of a cube: s³
• Compound interest: P(1 + r)ⁿ
• Inverse square law: F ∝ 1/r²

Multiplying Same Base: Add Exponents

Rule: aᵐ × aⁿ = aᵐ⁺ⁿ

Why? Count the factors.

2³ × 2⁴ = (2 × 2 × 2) × (2 × 2 × 2 × 2) = 2⁷

Three 2s times four 2s equals seven 2s.

Example: x⁵ × x³ = x⁸

You're combining factors, so add the exponent counts.

Dividing Same Base: Subtract Exponents

Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Why? Cancel common factors.

2⁵ ÷ 2³ = (2×2×2×2×2) / (2×2×2) = 2×2 = 2²

Five 2s divided by three 2s leaves two 2s.

Example: x⁷ / x⁴ = x³

Power of a Power: Multiply Exponents

Rule: (aᵐ)ⁿ = aᵐⁿ

Why? You're repeating the multiplication.

(2³)² = 2³ × 2³ = (2×2×2) × (2×2×2) = 2⁶

Two copies of three 2s equals six 2s.

Example: (x⁴)³ = x¹²

Power of a Product: Distribute the Exponent

Rule: (ab)ⁿ = aⁿbⁿ

Why? Everything in the parentheses gets multiplied n times.

(2 × 3)⁴ = (2×3)(2×3)(2×3)(2×3) = (2×2×2×2)(3×3×3×3) = 2⁴ × 3⁴

Example: (xy)³ = x³y³

Zero Exponent: Always 1

Rule: a⁰ = 1 (when a ≠ 0)

Why? Follow the pattern:

2³ = 8 2² = 4 2¹ = 2 2⁰ = ?

Each step divides by 2. So 2⁰ = 2 ÷ 2 = 1.

Or use the division rule: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰. But aⁿ ÷ aⁿ = 1. So a⁰ = 1.

Example: 100⁰ = 1, x⁰ = 1

Negative Exponents: Reciprocals

Rule: a⁻ⁿ = 1/aⁿ

Why? Continue the pattern:

2² = 4 2¹ = 2 2⁰ = 1 2⁻¹ = ?

Each step divides by 2. So 2⁻¹ = 1 ÷ 2 = 1/2.

2⁻² = 1/4 = 1/2² 2⁻³ = 1/8 = 1/2³

Negative exponents flip to the denominator.

Example: x⁻³ = 1/x³, (2/3)⁻² = (3/2)² = 9/4

Fractional Exponents: Roots

Rule: a^(1/n) = ⁿ√a (the nth root of a)

Why? Consider: what number, raised to the nth power, gives a?

If x = a^(1/2), then x² = a^(2/2) = a¹ = a. So x = √a.

a^(1/2) = √a (square root) a^(1/3) = ³√a (cube root) a^(1/4) = ⁴√a (fourth root)

Combining: a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)

8^(2/3) = (³√8)² = 2² = 4

Scientific Notation

Very large and very small numbers use exponents:

3,000,000 = 3 × 10⁶ 0.00005 = 5 × 10⁻⁵

The format is: a × 10ⁿ where 1 ≤ a < 10.

The exponent tells you how many places to move the decimal:

• Positive exponent: move right (bigger number)
• Negative exponent: move left (smaller number)

Speed of light: 3 × 10⁸ m/s Mass of electron: 9.1 × 10⁻³¹ kg

Why the Rules Work

All exponent rules follow from the basic definition: aⁿ means n copies of a multiplied together.

• Multiply: Combine the copies → add counts
• Divide: Cancel copies → subtract counts
• Power of power: Repeat the copies → multiply counts
• Zero exponent: No copies → just 1 (the multiplicative identity)
• Negative exponent: "Negative copies" = reciprocal

The rules aren't arbitrary — they're consequences of consistent counting.

Common Mistakes

Mistake: (a + b)² = a² + b² Reality: (a + b)² = a² + 2ab + b²

Exponents don't distribute over addition. (The distributive property is for multiplication over addition.)

Mistake: a⁻¹ = -a Reality: a⁻¹ = 1/a

Negative exponent means reciprocal, not negative number.

Mistake: x² × x³ = x⁶ Reality: x² × x³ = x⁵

Multiply same base = add exponents, not multiply.

Exponential Growth and Decay

Exponents model processes that grow or shrink by percentages:

Growth: A = P(1 + r)ᵗ (compound interest, population) Decay: A = P(1 - r)ᵗ (depreciation, radioactive decay)

Each time period multiplies by the same factor. That's why the exponent appears — it counts the multiplication periods.

The Core Insight

Exponents count factors.

3⁴ means four 3s multiplied. Every rule flows from this:

• Combining factors? Add the counts.
• Removing factors? Subtract the counts.
• No factors? You have 1.
• Negative factors? Go to the denominator.

The notation is compressed. The meaning is simple: how many times does the base appear in the multiplication?

Part 8 of the Algebra Fundamentals series.

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