Five simple rules turn towering, repeated multiplication into a single line of arithmetic — and explain why anything to the power of zero always equals 1.
An exponent is just shorthand for repeated multiplication — but once two exponent expressions need to be combined, five specific laws take over, and each one has its own exact rule for what to do with the exponents themselves.
An exponent (or power, or index) tells you how many times to multiply a number by itself. In 2⁴, the base is 2 and the exponent is 4, meaning 2 × 2 × 2 × 2.
2⁴ = 2 × 2 × 2 × 2 = 16. Read aloud as "two to the power of four" or "two to the fourth."
This is exactly the same idea already covered for squares (n²) and cubes (n³) in our Squares, Square Roots, Cubes & Cube Roots article — exponents simply extend that pattern to any power, not just 2 or 3.
Any non-zero number raised to the power of 0 equals exactly 1 — always, with no exceptions.
5⁰ = 1. 100⁰ = 1. Even 999,999⁰ = 1.
A negative exponent doesn't make the result negative — it means "take the reciprocal," flipping the number into a fraction.
2⁻³ = 1/2³ = 1/8 = 0.125. The result is a small positive fraction, not a negative number.
This is one of the most common exponent mistakes — assuming the negative sign carries into the final answer.
The fix: a negative exponent means "flip to a fraction," not "make the answer negative." 2⁻³ = 1/2³ = 1/8, a positive value less than 1.
When multiplying two powers with the same base, add the exponents.
2³ × 2⁴ = 2⁽³⁺⁴⁾ = 2⁷ = 128. Check directly: 8 × 16 = 128. ✓
When dividing two powers with the same base, subtract the exponents.
2⁵ ÷ 2² = 2⁽⁵⁻²⁾ = 2³ = 8. Check directly: 32 ÷ 4 = 8. ✓
When a power is itself raised to another power, multiply the exponents together.
(2³)² = 2⁽³ˣ²⁾ = 2⁶ = 64. Check directly: 8² = 64. ✓
This mixes up Law 1 (add, for multiplying same-base powers) with Law 3 (multiply, for a power raised to another power) — two different situations with two different rules.
The fix: 3² × 3³ = 3⁽²⁺³⁾ = 3⁵ = 243 — the exponents add because this is two separate powers being multiplied together, not one power raised to another. Multiplying exponents only applies to expressions like (3²)³, where a power itself gets raised to a further power.
When a product of two numbers is raised to a power, the exponent applies to each factor separately.
(2×3)³ = 2³ × 3³ = 8 × 27 = 216. Check directly: 6³ = 216. ✓
The same idea applies to division: the exponent applies separately to the numerator and denominator.
(6÷2)³ = 6³÷2³ = 216÷8 = 27. Check directly: 3³ = 27. ✓
This is a very common overreach — applying the power-of-a-product law to addition, where it simply doesn't hold.
The fix: the "distribute the exponent" laws only work for multiplication and division, never for addition or subtraction. (3+4)² = 7² = 49, but 3²+4² = 9+16 = 25 — completely different answers. There is no exponent law for the power of a sum; expanding (a+b)² properly requires multiplying it out as (a+b)×(a+b), a topic covered in algebra.
| Law | Rule |
|---|---|
| Product of Powers | aᵐ × aⁿ = aᵐ⁺ⁿ |
| Quotient of Powers | aᵐ ÷ aⁿ = aᵐ⁻ⁿ |
| Power of a Power | (aᵐ)ⁿ = aᵐˣⁿ |
| Power of a Product | (ab)ⁿ = aⁿbⁿ |
| Power of a Quotient | (a÷b)ⁿ = aⁿ÷bⁿ |
Tap any expression below to see which law applies and how it simplifies, step by step.
Tap an expression above to see the full breakdown.
Computer memory is measured in powers of 2, exactly as covered in our Number Base Systems article: 2¹⁰ = 1,024 bytes make a kilobyte, and 2²⁰ = 1,048,576 bytes make a megabyte. Bacteria populations that double every hour also grow by powers: starting from 1, after 10 hours there are 2¹⁰ = 1,024 bacteria.
A single grain of rice doubled on each square of a chessboard (1, 2, 4, 8, 16...) reaches over 18 quintillion grains by the 64th square — a classic illustration of just how explosively powers grow, sometimes called "the wheat and chessboard problem."
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Exponent laws are used constantly in scientific notation, compound growth calculations, computer science, and — very soon in your math journey — algebra, where these exact same five laws apply to expressions with letters instead of plain numbers. Getting comfortable with them now, using numbers you can double-check by hand, makes every later use of exponents dramatically easier.
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