Exponent rules are the building blocks of every algebra class — and the most-confused topic on the SAT and ACT math sections. The eight rules are simple individually, but they share enough surface similarity to trip up students who haven't internalized them. Here's the full cheat sheet with examples.
Rule 1: product rule (same base)
bm × bn = bm+n
When multiplying powers of the same base, add the exponents.
- 2³ × 2⁴ = 2⁷ = 128
- x² × x⁵ = x⁷
- 10⁶ × 10⁻² = 10⁴ = 10,000
Common mistake: applying this rule across different bases. 2³ × 3² has no shortcut — calculate each separately and multiply.
Rule 2: quotient rule (same base)
bm ÷ bn = bm−n
- 2⁷ ÷ 2³ = 2⁴ = 16
- x⁵ ÷ x² = x³
- 10³ ÷ 10⁵ = 10⁻² = 0.01
Rule 3: power of a power
(bm)n = bmn
- (2³)² = 2⁶ = 64
- (x⁴)³ = x¹²
Critical distinction: (2³)² ≠ 2³ × 2² = 32. Multiply, don't add. The whole expression 2³ is being squared.
Rule 4: power of a product
(ab)n = anbn
- (2x)³ = 2³x³ = 8x³
- (3y)² = 9y²
The exponent distributes over multiplication. It does NOT distribute over addition — (a + b)² ≠ a² + b². The correct expansion is a² + 2ab + b².
Rule 5: power of a quotient
(a/b)n = an/bn
- (2/3)³ = 8/27
- (x/y)⁵ = x⁵/y⁵
Rule 6: zero exponent
b0 = 1 (for any nonzero b)
This follows from the quotient rule: bm / bm = bm−m = b0. Anything divided by itself is 1.
- 2⁰ = 1
- (−5)⁰ = 1
- x⁰ = 1 (assuming x ≠ 0)
- 0⁰ is conventionally 1, though debated
Rule 7: negative exponent
b−n = 1/bn
A negative exponent means take the reciprocal:
- 2⁻³ = 1/8 = 0.125
- x⁻² = 1/x²
- (2/3)⁻¹ = 3/2 (flips the fraction)
Negative exponents do not mean negative numbers. 2⁻³ is positive 0.125, not negative anything.
Rule 8: fractional exponent
b1/n = ⁿ√b and bm/n = ⁿ√(bm)
- 161/2 = √16 = 4
- 271/3 = ∛27 = 3
- 82/3 = ∛(8²) = ∛64 = 4 (or equivalently (∛8)² = 2² = 4)
Combining the rules
The hard problems chain multiple rules together:
Simplify (2x²y⁻³)³.
- Apply Rule 4 (power of product): 2³ × (x²)³ × (y⁻³)³
- Apply Rule 3 (power of power): 8 × x⁶ × y⁻⁹
- Apply Rule 7 (negative exponent): 8x⁶ / y⁹
Order of operations matters
−2² is −4, not 4. The exponent applies only to 2. To square negative 2, write (−2)² = 4. The minus sign is not part of the base unless explicitly inside parentheses.
This is the #1 source of sign errors on standardized tests. Always parenthesize negative bases.
The fundamental identity
One identity ties everything together: bm × bn = bm+n. From this single equation, every other rule can be derived — the quotient rule, zero-exponent rule, negative-exponent rule, fractional-exponent rule. Memorizing eight rules separately is unnecessary; understanding why the product rule forces all the others is the deeper learning.
Scientific notation: the cleanest application
Scientific notation writes any number as a × 10n, where 1 ≤ a < 10. Multiplication and division of large/small numbers becomes trivial:
- (3 × 10⁴) × (2 × 10⁻⁶) = 6 × 10⁻² = 0.06 (multiply coefficients, add exponents)
- (8 × 10⁹) ÷ (4 × 10⁵) = 2 × 10⁴ = 20,000 (divide coefficients, subtract exponents)
Used everywhere in chemistry (Avogadro's number, 6.022 × 10²³), physics (speed of light, 3 × 10⁸ m/s), and any context with extreme magnitudes. The exponent rules are exactly what makes scientific notation work.
Common test traps
Three patterns that show up repeatedly on the SAT, ACT, and AP exams:
- Distribution over addition. (a + b)² ≠ a² + b². Expand: a² + 2ab + b². The cross term 2ab is the FOIL leftover.
- Negative bases vs negative exponents. (−2)² = 4. −2² = −4. The first squares −2; the second squares 2 then negates.
- Same-base trap. 2³ × 8² isn't directly combinable until you rewrite 8 as 2³, giving 2³ × (2³)² = 2³ × 2⁶ = 2⁹. Always check whether different bases share a common base.
Practice with the tool
Our exponent calculator evaluates any base raised to any exponent — positive, negative, fractional, decimal — and shows the answer in scientific notation when it gets large. Useful for verification when you've simplified an expression with multiple rules and need to confirm the result.