Exponent Rules (Laws of Exponents) With Examples

The laws of exponents combine powers that share a base. Multiply two powers by adding the exponents (a^m * a^n = a^(m+n)), divide by subtracting (a^m / a^n = a^(m-n)), and raise a power to a power by multiplying ((a^m)^n = a^(m*n)). Each rule follows directly from what an exponent counts.

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What the rules do

An exponent counts how many times a base is written as a factor, so combining powers just means tracking that count. You never have to expand a power to combine it with another. Multiplying two powers of the same base stacks the factors end to end, and the exponent records the new total.

The seven core rules, each stated once with a worked check. Every rule assumes the base is nonzero wherever division or a zero exponent is involved.

Multiply and divide same-base powers

Multiplying adds the exponents. When the bases match, lining up both sets of factors gives one longer string of the same factor, so you add the two counts. This only works when the bases match. A product of powers with different bases has nothing to combine, because the factors are different.

Dividing subtracts the exponents. Each factor in the denominator cancels one matching factor in the numerator, so you remove that many from the count. What remains is the difference of the two exponents.

Raise a power to a power

Stacking powers multiplies the exponents. Raising a power to another power uses the inner power as a factor several times, and the product rule adds the inner exponent that many times. Adding it repeatedly is the same as multiplying. Do not confuse this with the product rule: a power of a power multiplies, while a plain product of powers adds.

Distribute a power over a product

A power spreads across multiplication. When the whole product is raised to a power, every factor inside picks up that exponent. Regrouping the repeated factors shows each base carrying the same power. The same distribution works over division, sending the exponent onto both the numerator and the denominator.

The exponent does not distribute over addition. A squared sum is not the sum of the squares, because the middle cross term is missing.

Zero, negative, and fractional exponents

A zero exponent always gives one. Dividing any power by itself leaves the difference of equal exponents, which is zero, and a nonzero quantity divided by itself is one. So a zero exponent is what keeps the quotient rule consistent. The zero base raised to the zero power is left undefined.

A negative exponent means the reciprocal. Applying the quotient rule to a small power over a larger one produces a negative exponent, and computing the same fraction directly gives the reciprocal of the positive power. A negative exponent never makes a number negative. It flips the power into a fraction. Moving a factor across the fraction bar flips the sign of its exponent.

A unit fraction in the exponent is a root. The denominator names the root, so a one-over-n exponent is the n-th root of the base. The power rule forces this, since raising that root back to the n-th power returns the base. For a general fraction, the numerator is a power and the denominator is a root.

Different bases share none of these shortcuts, and exponents never distribute over addition. Track the count of factors and the rule for each operation follows on its own.