Exponent Calculator

How it works

An exponent says how many times to multiply the base by itself: 2^10 is ten 2s multiplied together, 1024. The same rule extends naturally beyond counting numbers. A negative exponent is a reciprocal, 2^-3 = 1 ÷ 2^3 = 0.125. A fractional exponent is a root: a^(1/q) is the q-th root of a, and a^(p/q) is that root raised to the p-th power, so 8^(2/3) is the cube root of 8 squared, 2^2 = 4.

Enter a fractional exponent as its numerator and denominator rather than as a decimal, so 2/3 stays exactly 2/3 instead of 0.667. The calculator takes the root first and then the power, which is also the easiest order by hand because the root of a perfect power is a whole number.

The other three modes apply the laws of exponents to two powers of the same base. Multiplying adds the exponents, dividing subtracts them, and raising a power to a power multiplies them. The Simplified form shows the single power these laws produce, and the Value evaluates it.

A negative base only has a real answer for a whole-number exponent. (-8)^(1/3) could be the real cube root -2, but (-8)^(2/6) is the same exponent and gives -2 if you reduce the fraction first (the cube root of -8) but 2 if you square first (the sixth root of 64), so the fractional case is reported as not real. Use the root calculator for the real odd root of a negative number.

Formula

a^n  = a × a × … × a   (n factors)         a^0 = 1
a^(-n) = 1 ÷ a^n
a^(p/q) = (q-th root of a)^p
a^m × a^n = a^(m + n)
a^m ÷ a^n = a^(m − n)
(a^m)^n   = a^(m × n)

Example

2^10 = 1024, the number of bytes in a kibibyte. With the exponent -3 instead, 2^-3 = 1 ÷ 2^3 = 1 ÷ 8 = 0.125.

343^(2/3): the cube root of 343 is 7, and 7^2 = 49.

Product mode with base 2 and exponents 3 and 4: 2^3 × 2^4 = 2^(3 + 4) = 2^7 = 128. Quotient mode with base 5 and exponents 6 and 4: 5^6 ÷ 5^4 = 5^2 = 25. Power of a power with base 3 and exponents 2 and 3: (3^2)^3 = 3^6 = 729.

Assumptions and limitations

• 0^0 is reported as 1, the usual convention in algebra and in most calculators; some contexts leave it undefined.
• A negative base with a fractional exponent is reported as having no real result, even when the denominator is odd. The root calculator gives the real cube root of a negative number.
• Values are cleaned to 15 significant digits to remove floating-point noise, and shown to 6 decimal places. Results above about 1.8 × 10^308 cannot be represented and are reported as too large.
• In the three same-base modes the exponents may be any numbers, including negative or decimal ones; the laws of exponents hold for all of them as long as the base is positive, or the resulting exponent is a whole number.

Frequently asked questions