Exponent Rules: The Laws of Exponents With Examples

Exponent rules tell you how to combine powers without expanding them. The key ones are: multiply powers with the same base by adding exponents (x³ · x⁴ = x⁷), divide by subtracting (x⁶ ÷ x⁴ = x²), and raise a power to a power by multiplying ((x²)³ = x⁶). Zero, negative and fractional exponents follow from those three: x⁰ = 1, x⁻ⁿ = 1/xⁿ and x^1/n = ⁿ√x.

Every rule comes from the meaning of an exponent as repeated multiplication. x³ is x · x · x, and x⁴ is x · x · x · x. Multiply them and you have seven x’s, which is why the exponents add. If you forget a rule, writing out a small example like this will rebuild it in seconds.

All the exponent rules

The product and quotient rules require the same base. 2³ · 3⁴ cannot be simplified to a single power; you can only evaluate it: 8 × 81 = 648.

The three core rules

Product rule

2³ · 2⁴ means (2 · 2 · 2)(2 · 2 · 2 · 2), seven 2s in all, so it equals 2⁷ = 128.

Quotient rule

In 5⁶ ÷ 5⁴, four of the six 5s on top cancel with the four on the bottom, leaving 5² = 25.

Power rule

(3²)³ is 3² written three times: 3² · 3² · 3² = 3⁶ = 729. Applied to a product, every factor inside the parentheses gets the exponent, including the number: (2x³)⁴ = 2⁴ · x¹² = 16x¹².

Zero and negative exponents

Extend the quotient rule to equal exponents: x³ ÷ x³ = x⁰. But anything divided by itself is 1, so x⁰ = 1. Extend it further, to a smaller exponent on top: x² ÷ x⁵ = x⁻³, and canceling directly gives 1/x³. So:

• 4⁻² = 1/4² = 1/16
• 10⁻³ = 1/1,000 = 0.001, which is how scientific notation writes small numbers
• (1/2)⁻³ = 2³ = 8: a negative exponent on a fraction flips it

A negative exponent moves a factor across the fraction bar. In 3x⁻² the exponent applies only to x, so 3x⁻² = 3/x². In (3x)⁻² it applies to both: 1/(9x²).

What about 0⁰? It is left undefined in some contexts and defined as 1 in others, including most of algebra, combinatorics and programming languages. The rule x⁰ = 1 is normally stated for x ≠ 0 to sidestep the question.

Fractional exponents and roots

A fractional exponent combines a root and a power. The denominator is the root; the numerator is the power.

• 8^2/3 = (∛8)² = 2² = 4
• 16^3/4 = (⁴√16)³ = 2³ = 8
• 27^−1/3 = 1/∛27 = 1/3
• 32^0.4 = 32^2/5 = (⁵√32)² = 2² = 4

Take the root first when you can, because it keeps the numbers small. The fraction exponents calculator shows both orders, and the nth root calculator evaluates any root. Radicals have their own simplification techniques, covered in how to simplify square roots.

Negative bases

The sign of a power of a negative number depends on whether the exponent is even or odd. An even number of negative factors pairs up into positives, and an odd number leaves one negative behind:

• (−3)² = 9 and (−3)⁴ = 81: even exponents give positive results
• (−3)³ = −27 and (−3)⁵ = −243: odd exponents keep the minus sign

Fractional exponents of negative numbers need care. (−8)^1/3 = −2, because (−2)³ = −8, so odd roots of negatives are real. But (−4)^1/2 = √−4 has no real value, since no real number squares to a negative. Calculators and spreadsheets often return an error for any fractional power of a negative base, even when an odd root exists, so take the root of the positive value and restore the sign yourself.

Simplifying expressions step by step

Combine the rules in order: deal with parentheses using the power rules, then combine like bases, then rewrite negative exponents as fractions.

Common mistakes

The last row is worth remembering: adding equal powers doubles them, which adds just 1 to the exponent. Exponents grow quickly under multiplication, not addition.

Exponents in the real world

Compound growth. Money growing at 5% a year for 10 years is multiplied by 1.05¹⁰ ≈ 1.6289, so $1,000 becomes about $1,628.89. Halving. Something that halves three times is multiplied by (1/2)³ = 1/8. Computer memory. Sizes come in powers of 2: 2¹⁰ = 1,024, which is why a kibibyte has 1,024 bytes, as explained in data storage units explained.

When the unknown is the exponent itself, as in 1.05ᵗ = 2, you need logarithms, the inverse of exponents. See logarithm rules or the solve for exponent calculator. For direct evaluation, the exponent calculator handles any base and power, and the large exponents calculator gives exact results for huge numbers like 3¹⁰⁰.