Exponent Rules Calculator

The five laws of exponents in one table

An exponent is a shorthand for repeated multiplication: x^4 means x times x times x times x. Five laws let you rewrite any such expression without multiplying everything out, and they are listed in the OpenStax College Algebra chapter on exponents and collected on the Wolfram MathWorld page for exponent laws. The table below is the whole toolkit, each rule shown in symbols and with real numbers.

Every rule carries one condition: the base has to be the same on both powers before you combine them. That single condition causes most of the wrong answers on homework, because 2^3 x 3^2 looks like it should collapse but never does. The sections below work each rule out with plain numbers so the pattern is visible.

Type a base and two exponents into the calculator above and it applies all five laws at once, printing both the symbolic rewrite and the checked arithmetic for each one.

Product rule: add the exponents

The product rule says that multiplying two powers of the same base adds the exponents. Take 2^3 x 2^4. The long way, 2^3 is 8 and 2^4 is 16, and 8 x 16 gives 128. The short way, the exponents add: 3 + 4 = 7, and 2^7 is 128. Both roads land on the same number, which is the whole point of the law: it is not an approximation, it is a restatement of what repeated multiplication does.

Counting the twos shows why. 2^3 contributes three factors of 2 and 2^4 contributes four more, so the product holds seven factors of 2 in a row, which is the definition of 2^7. Decimals follow the same rule: 1.5^2 x 1.5^3 is 1.5^5, and 1.5^5 = 7.59375. Check it the long way and 2.25 x 3.375 gives 7.59375 again.

The rule survives awkward signs too. 5^-2 x 5^4 = 5^2 = 25, because the exponents -2 and 4 add to 2. The base never has to be bigger than 1, integer, or positive; only sameness matters.

Quotient rule and negative exponents

Division undoes multiplication, so the quotient rule subtracts exponents. 3^5 / 3^2 removes two of the five threes and leaves 3^3, which is 27. The long way agrees: 243 / 9 = 27. When the bottom exponent is larger, the result goes negative: 10^3 / 10^7 = 10^-4, and 10^-4 = 0.0001. That is 1000 divided by 10,000,000, and the subtraction 3 - 7 = -4 keeps track of how far the decimal point travels.

A negative exponent means reciprocal, nothing more. 2^-3 is 1 / 2^3, which is 1/8, or 0.125. Likewise 4^-2 = 1/16 = 0.0625. Flipping the sign of the exponent flips the power across the fraction bar, so (3/4)^-2 can be read as (4/3)^2 = 16/9, about 1.7778. Money grows out of this rule directly: a 4x return spread over two doublings is 2^2, and a half return per period shows up as a base smaller than 1.

The one place the rule refuses to run is a zero base with a nonpositive exponent. 0 / 0 has no defined value, so 0^-2 stays undefined no matter what a calculator keypad suggests.

Power of a power, and powers of products

When a power is itself raised to a power, the exponents multiply. (5^2)^3 means (5^2) x (5^2) x (5^2), and each 5^2 holds two fives, so the whole thing holds six: 5^6 = 15625. Computing the inside first gives the same number the long way, since 5^2 = 25 and 25^3 = 15625. The multiplication 2 x 3 = 6 is the shortcut.

A power over a product splits into a product of powers: (x y)^a = x^a y^a. So (2 x 3)^3 = 2^3 x 3^3 = 8 x 27 = 216, which matches 6^3 = 216. That split is the step people skip when they square a sum, and skipping it there is an error, because a sum is not a product. Powers of quotients split the same way: (8/2)^3 = 4^3 = 64, and also 8^3 / 2^3 = 512/8 = 64.

Chained powers run from the top down. 2^3^2 is read as 2^(3^2) = 2^9 = 512 in standard notation, not (2^3)^2 = 64. Parentheses decide it, so when in doubt, write them.

Zero exponents and the zero base

Anything (except zero) to the zero power is 1. The quotient rule shows why without any hand waving: take x^a / x^a. Any number divided by itself is 1, and the rule says the same fraction is x^(a - a) = x^0, so x^0 has to equal 1. Try it: 9^0 = 1, 0.37^0 = 1, and (-6)^0 = 1.

The zero base is the exception that proves the rule. 0^3 / 0^3 asks for 0/0, which is undefined, so 0^0 gets no value from this argument. Some fields define 0^0 = 1 by convention for counting empty products, but on a homework sheet the safe answer is undefined unless the class says otherwise. Negative exponents on a zero base, like 0^-2, are undefined for the same reason: they ask for division by zero.

Watch the placement of a minus sign as well. -9^0 means -(9^0), which is -1, while (-9)^0 = 1. The exponent binds to the 9 before the leading minus is applied.

Five moves that break the rules

Most lost points come from applying a law across two different bases or across an addition. The table below lists the moves that look legal and are not, next to the correct value.

One habit catches all five: before touching exponents, ask whether the operation between the powers is a product, a quotient, or neither. Only products and quotients of the same base have a law. Sums need to be evaluated or factored first, and mixed bases stay put.

When a problem does mix bases, look for a hidden rewrite. 4^5 x 2^3 becomes (2^2)^5 x 2^3 = 2^10 x 2^3 = 2^13 = 8192, because 4 is 2 squared. Rewriting to a shared base is the standard trick behind half the exponent questions on standardized tests.

Check every law with numbers you can verify

Each law earns its keep when the arithmetic on both sides lands on the same number. Work the left column by hand, then check the right column with a calculator, and the rules stop being memorized strings and start being facts you can see.

Two of these rows deserve a second look. The row for 4^3 / 4^3 shows why the zero exponent is defined the way it is: dividing any nonzero number by itself gives 1, so x^0 has to equal 1 for the quotient rule to hold. The negative exponent row shows the same logic in reverse, since 3^2 / 3^4 = 3^-2 and also equals 1/9.