Enter values into any two of the input fields to solve for the third.
What is an exponent?
Exponentiation is a mathematical operation, written as an, involving the base a and an exponent n. In the case where n is a positive integer, exponentiation corresponds to repeated multiplication of the base, n times.
an = a × a × ... × a
n times
This tool can handle bases that are negative, yet it is unable to produce results involving imaginary numbers. Fractions themselves are not supported, although you may calculate fractional powers provided you enter the exponent as a decimal.
Basic exponent laws and rules
When exponents that share the same base are multiplied, the exponents are added.
Similarly, when divided bases are raised to an exponent, the exponent is distributed to both bases.
(
a
b
)n
=
an
bn
EX: (
2
5
)2
=
2
5
×
2
5
=
4
25
(
2
5
)2
=
22
52
=
4
25
When an exponent is 1, the base remains the same.
a1 = a
If the exponent equals zero, raising any base yields 1, even though the case 00 sparks discussion over whether it should be considered 1 or left undefined. In most practical contexts, treating 00 as 1 simplifies matters.
a0 = 1
Shown below is an example of an argument for a0=1 using one of the previously mentioned exponent laws.
If an × am = a(n+m)
Then an × a0 = a(n+0) = an
Thus, the only way for an to remain unchanged by multiplication, and this exponent law to remain true, is for a0 to be 1.
A fractional exponent whose numerator is 1 corresponds to extracting the nth root of the base. Below we illustrate a case where the numerator differs from 1, applying both the displayed rule and the multiplication rule for like bases introduced earlier. Remember that the calculator handles fractional powers, but you need to type the exponent as a decimal.
Exponents with negative bases can also be evaluated, following principles similar to those for positive bases. When a negative base is raised to an even positive integer, the magnitude matches that of the corresponding positive base and the sign is positive; with an odd positive integer exponent, the magnitude stays the same but the result becomes negative. Fractional exponents on negative bases obey the same formulas, yet they lead to imaginary numbers because extracting roots of a negative value is not possible in the real number system. The example below demonstrates this, but note that the calculator cannot handle imaginary results—any such input will produce "NAN" (not a number). In essence, the numeric outcome mirrors that of a positive base, merely flagged as imaginary.
