Exponent Calculator
Enter values into any two of the input fields to solve for the third.
What is an exponent?
Exponentiation is the operation written as an: a base a raised to an exponent n. When n is a positive integer, it is simply repeated multiplication of the base, n times:
an = a × a × … × a (n times)
This calculator accepts negative bases, but it does not compute imaginary numbers. It also does not accept fractions as input, although fractional exponents work fine when entered in decimal form — 20.5, for example, is the square root of 2.
Basic exponent laws and rules
Multiplying like bases adds the exponents: an × am = a(n+m). Example: 22 × 24 = 4 × 16 = 64, and 2(2+4) = 26 = 64.
A negative exponent reciprocates the base: a(−n) = 1an. Example: 2(−3) = 1 ÷ 2 ÷ 2 ÷ 2 = 18 = 0.125.
Dividing like bases subtracts the exponents: aman = a(m−n). Example: 2224 = 416 = 14, and 2(2−4) = 2−2 = 14.
Raising a power to a power multiplies the exponents: (am)n = a(m×n). Example: (22)4 = 44 = 256, and 2(2×4) = 28 = 256.
An exponent distributes over multiplication and division: (a × b)n = an × bn, and (ab)n = anbn. Example: (2 × 4)2 = 82 = 64 = 22 × 42 = 4 × 16.
Exponent 1 and 0: a1 = a, and a0 = 1 for any base. The zero rule follows from the product law: since an × a0 = a(n+0) = an, the factor a0 must equal 1 for the value to stay unchanged. Whether 00 should be 1 or undefined is debated; defining it as 1, as this calculator does, is the convenient choice in most applications.
Fractional exponents are roots: an exponent of 1n takes the nth root of the base, and other fractions combine a power with a root — 82/3, entered as 80.666666666667, is (∛8)2 = 4.
Negative bases
Negative bases follow the same laws. Raised to a positive integer, the magnitude matches the positive base: an even exponent gives a positive result ((−2)2 = 4) and an odd exponent a negative one ((−2)3 = −8). Fractional exponents on negative bases are different: they require roots of negative numbers, which are imaginary, so this calculator reports them as unsolvable rather than returning a real answer.
Frequently Asked Questions
How do I solve for the exponent?
Enter the base and the result and leave the exponent blank. The calculator uses logarithms: exponent = ln(result) / ln(base). This needs positive values, and the base cannot be 1, since 1 raised to any power is 1.
How do I solve for the base?
Enter the exponent and the result and leave the base blank. The base is the corresponding root of the result: base = result(1/exponent). For example, a result of 32 with an exponent of 5 gives base = 32(1/5) = 2.
Can the base be e?
Yes. Type the letter e as the base (or use the "use e as base" link) to calculate powers of Euler's number, approximately 2.718281828459. For example, e2 = 7.3890560989307.
What is 0 to the power of 0?
It is a debated special case. This calculator follows the common convention and returns 1, which keeps the exponent laws consistent: an × a0 = a(n+0) can only hold for every a if a0 = 1.
Why can't a negative base have a fractional exponent here?
A fractional exponent means taking a root, and even roots of negative numbers are not real numbers — they are imaginary. This calculator works with real numbers only, so such inputs report that they cannot be solved.