Skill: exponents

Exponent Rules

An exponent is shorthand for repeated multiplication: 2⁴ means 2·2·2·2 = 16.
Five small rules run everything — multiplying powers, dividing powers, powers of powers, zero, and negative
exponents. Learn them here and you can simplify almost any power expression.

1 Understand: What an Exponent Is

The big number is the base; the small raised number is the exponent —
it counts how many times the base multiplies itself.

2⁴ = 2 · 2 · 2 · 2 = 16

The five rules

Multiply powers, add exponents: x³ · x⁴ = x⁷
Divide powers, subtract exponents: 2⁵ / 2² = 2³
Power of a power, multiply exponents: (3²)⁴ = 3⁸
Zero exponent: anything nonzero to the 0 power is 1 — 5⁰ = 1
Negative exponent: flip it into the denominator — 2⁻³ = 1/2³ = 1/8

Why it matters

Exponents are the language of growth — compound interest, populations, computer memory (2¹⁰ = 1024 bytes) —
and the foundation of polynomials, scientific notation, and exponential functions.

Where it’s used

Compound interest · computer storage · scientific measurements · area and volume formulas (x², x³).

FiboMap node: alg.exponent-rules · CCSS 8.EE.1 · Know first: multiplication facts

2 See It

Exponents just count factors. Watch what the notation is really saying.

Exponents just count factors. x³ · x² has 3 + 2 = 5 factors of x — that is why you add exponents when multiplying.

Doubling every hour is 5·2ⁿ. Exponents turn “repeated doubling” into one short expression.

3 Worked Examples

Follow each step. Name the rule first, then apply it.

Example 1 Multiply: x³ · x⁴

Thinking: same base, multiplying → add the exponents.
x³ · x⁴ = x³⁺⁴ = x⁷.
Check: 3 factors times 4 factors = 7 factors of x.

x³ · x⁴ = x⁷

Example 2 Divide: 2⁵ / 2²

Thinking: same base, dividing → subtract the exponents.
2⁵ / 2² = 2⁵⁻² = 2³ = 8.
Check: 32 / 4 = 8.

2⁵ / 2² = 8

Example 3 Power of a power: (3²)⁴

Thinking: a power raised to a power → multiply the exponents.
(3²)⁴ = 3^2×4 = 3⁸ = 6561.
Check: 3⁴ = 81 and 81² = 6561.

(3²)⁴ = 6561

Example 4 Zero and negative: 5⁰ and 2⁻³

Thinking: anything nonzero to the 0 power is 1; a negative exponent flips to the denominator.
5⁰ = 1; 2⁻³ = 1/2³ = 1/8.
Check: the pattern 2³=8, 2²=4, 2¹=2, 2⁰=1, 2⁻¹=1/2 keeps halving — consistent.

5⁰ = 1 · 2⁻³ = 1/8

4 Common Mistakes

Three errors show up on almost every exponents quiz. Spot them now and they will never cost you points.

Mistake 1: (x²)³ = x⁵

Wrong

Adding exponents for a power of a power: 2 + 3 = 5.
The outer exponent hits every inner factor, so adding undercounts.

Right

(x²)³ = x^2·3 = x⁶
Multiply the exponents: 2 × 3 = 6.

Rule: “Power of a power → multiply.” The outer exponent hits every inner factor.

Mistake 2: 2³ = 6

Wrong

Multiplying base × exponent: 2 × 3 = 6.
The exponent counts multiplications, not additions.

Right

2³ = 2 · 2 · 2 = 8
Three factors of 2, multiplied.

Memory hook: 2³ is “2, three times, multiplied” — 2·2·2 — not 2·3.

Mistake 3: x⁴ · x³ = x¹²

Wrong

Multiplying exponents when the rule says add: 4 × 3 = 12.
That mixes up two different rules.

Right

x⁴ · x³ = x⁴⁺³ = x⁷
Four factors plus three factors = seven factors.

Chant the trio: “Multiply → add. Divide → subtract. Power of power → multiply.”

5 Quick Check

Try each one on paper first, then reveal the answer.

1. Simplify: 4³ · 4². Leave your answer as a power.

Answer

Same base, multiplying → add: 4³ · 4² = 4³⁺² = 4⁵ (= 1024).

2. Evaluate: 10³ / 10¹.

Answer

Same base, dividing → subtract: 10³⁻¹ = 10² = 100.

3. Evaluate: (2³)².

Answer

Power of a power → multiply: 2^3·2 = 2⁶ = 64.

Key Points to Remember

x^m · xⁿ = x^m+n — multiply → add
x^m / xⁿ = x^m−n — divide → subtract
(x^m)ⁿ = x^m·n — power of a power → multiply
x⁰ = 1 (for x ≠ 0); x⁻ⁿ = 1xⁿ
2³ = 2·2·2 = 8 — never 2·3

Practice exponent problems
Try the exponent tool
See exponents on FiboMap