TL;DR:
An exponent counts how many times the base is used as a factor.
Multiplying adds the counts, dividing subtracts them, and a power of a power multiplies them.
Zero and negative exponents are the values that keep those rules working.
Exponent rules look like a list of arbitrary laws until you remember what an exponent is: a count of factors. Once you count, each rule is something you can see, and you stop needing to memorize which sign goes where.
An exponent is a count of factors
2⁴ means four 2s multiplied: 2 × 2 × 2 × 2 = 16. The 2 is the base and the 4 is the exponent. Written out like this, the rules below are only a matter of counting how many 2s there are.
Worked example: 2³ × 2⁴ = 2⁷
Write each power out: 2³ = 2 × 2 × 2 and 2⁴ = 2 × 2 × 2 × 2. Multiplying them places the factors side by side: (2 × 2 × 2) × (2 × 2 × 2 × 2). Count the 2s: three plus four makes seven, so the result is 2⁷.
Check with numbers. 2³ = 8, 2⁴ = 16, and 8 × 16 = 128. Also 2⁷ = 128, since doubling from 2 gives 4, 8, 16, 32, 64, 128. They agree.
The general rule is aᵐ × aⁿ = aᵐ⁺ⁿ. It requires the same base, because only then are you counting the same kind of factor.
Quotients, powers of powers, and products
Quotient.
2⁷ ÷ 2³cancels three of the seven 2s, leaving four:2⁴ = 16. Check:128 ÷ 8 = 16. The rule isaᵐ ÷ aⁿ = aᵐ⁻ⁿ.Power of a power.
(2³)⁴means four copies of2³, so there are3 × 4 = 12factors of 2:2¹². Check:(2³)⁴ = 8⁴ = 8 × 8 × 8 × 8 = 64 × 64 = 4096, and2¹² = 4096. The rule is(aᵐ)ⁿ = aᵐⁿ.Power of a product.
(2 × 3)³ = 6³ = 216, and2³ × 3³ = 8 × 27 = 216. The rule is(ab)ⁿ = aⁿbⁿ.
Where zero and negative exponents come from
Counting factors cannot make sense of "zero factors" directly, so we ask what value keeps the quotient rule true. Look at the pattern of powers of 2, each one half the previous as the exponent falls by one.
Power | Value |
|---|---|
2³ | 8 |
2² | 4 |
2¹ | 2 |
2⁰ | 1 |
2⁻¹ | 1/2 |
2⁻² | 1/4 |
The quotient rule agrees: 2³ ÷ 2³ = 2⁰, and any number divided by itself is 1, so 2⁰ = 1. Likewise 2² ÷ 2³ = 2⁻¹, and by cancelling, 4 ÷ 8 = 1/2. So a negative exponent means a reciprocal: a⁻ⁿ = 1/aⁿ.
The same demand gives fractional exponents. If (9^½)² = 9^(½·2) = 9¹ = 9, then 9^½ must be a number whose square is 9, which is 3. So an exponent of one half means a square root.
Putting several rules together
Simplify (x³y²)⁴ ÷ (x⁵y). Apply the power rule to the top: x¹²y⁸. Then divide, subtracting exponents base by base: x¹²⁻⁵ = x⁷ and y⁸⁻¹ = y⁷. The result is x⁷y⁷.
Test with numbers. Let x = 2 and y = 1: the original is (8)⁴ ÷ 32 = 4096 ÷ 32 = 128, and x⁷y⁷ = 2⁷ = 128. Now x = 1, y = 2: the original is (4)⁴ ÷ 2 = 256 ÷ 2 = 128, and x⁷y⁷ = 2⁷ = 128. Testing a couple of simple values catches most slips.
Why scientists care about the rules
Scientific notation writes a number as something times a power of ten. Multiply 3 × 10⁴ by 2 × 10³: multiply the front numbers, 3 × 2 = 6, and add the exponents, 4 + 3 = 7, giving 6 × 10⁷. In ordinary numbers, 30,000 × 2,000 = 60,000,000, which is 6 × 10⁷. The product rule is doing the counting of zeros for you.
Negative exponents handle small numbers the same way: 10⁻³ = 1/1000 = 0.001. Moving the exponent down by one moves the decimal point one place to the right, matching the halving pattern in the table, only with tenths instead of halves.
How to check your answer
Evaluate both sides with small whole numbers, as with 128 and 4096 above.
If the answer should be bigger than the input, a negative exponent should have made it smaller. Sanity-check the direction.
For rules that need a common base, confirm the bases really match before adding exponents.
Common mistakes
Adding exponents when the bases differ.
2³ × 3⁴is not6⁷.Adding powers instead of multiplying them.
2³ + 2⁴ = 8 + 16 = 24, which is not2⁷ = 128. The rules apply to products, not sums.Writing
(a + b)² = a² + b². For example,(1 + 2)² = 9, but1² + 2² = 5.Mixing up
−3²and(−3)². The first is−(3 × 3) = −9, the second is(−3)(−3) = 9.Treating a negative exponent as a negative number.
2⁻³is1/8, which is positive.
Where you will use it next
These rules let you compare very large and very small numbers in science and are the base for exponential growth. They also lead to the idea that a logarithm asks which exponent you need, taken up in logarithms as questions about exponents. Functions with a variable exponent, such as f(x) = 2ˣ, follow the same laws; for that, see what a function really is.
Frequently asked questions
Why is anything to the power zero equal to 1?
Because aⁿ ÷ aⁿ must be 1 and, by the quotient rule, it is also a⁰. The only exception is 0⁰, which is left undefined in most courses.
Do these rules work for non-whole exponents?
Yes. The rules were chosen so that they keep working, which is how fractional and negative exponents get their meaning.
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