Exponent Rules

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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.

2 · 2 · 2 · 2 = 24 expanded: four factors of 2 2 · 2 · 2 · 2 = 16 x3 · x2 = x5 3 factors + 2 factors = 5 factors
Exponents just count factors. x3 · x2 has 3 + 2 = 5 factors of x — that is why you add exponents when multiplying.
Doubling every hour: 5 · 2n Hour Bacteria 05 110 220 340 480 ×2 ×2 ×2 ×2 Each hour multiplies by 2 — the exponent counts the doublings.
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
  1. Thinking: same base, multiplying → add the exponents.
  2. x3 · x4 = x3+4 = x7.
  3. Check: 3 factors times 4 factors = 7 factors of x.
x3 · x4 = x7
Example 2 Divide: 25 / 22
  1. Thinking: same base, dividing → subtract the exponents.
  2. 25 / 22 = 25−2 = 23 = 8.
  3. Check: 32 / 4 = 8.
25 / 22 = 8
Example 3 Power of a power: (32)4
  1. Thinking: a power raised to a power → multiply the exponents.
  2. (32)4 = 32×4 = 38 = 6561.
  3. Check: 34 = 81 and 812 = 6561.
(32)4 = 6561
Example 4 Zero and negative: 50 and 2−3
  1. Thinking: anything nonzero to the 0 power is 1; a negative exponent flips to the denominator.
  2. 50 = 1;  2−3 = 1/23 = 1/8.
  3. 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 = x2·3 = 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: 23·2 = 26 = 64.

Key Points to Remember

  • xm · xn = xm+n — multiply → add
  • xm / xn = xm−n — divide → subtract
  • (xm)n = xm·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
Fibo · Learn, Practice & Explore Math
Fibo · Learn, Practice & Explore Math