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 a^{n}, 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.
a^{n} = a × a × ... × a
n times
The calculator above accepts negative bases, but does not compute imaginary numbers. It also does not
accept fractions, but can be used to compute fractional exponents, as long as the exponents are input in
their decimal form.
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}
=
a^{n}
b^{n}
EX: (
2
5
)^{2}
=
2
5
×
2
5
=
4
25
(
2
5
)^{2}
=
2^{2}
5^{2}
=
4
25
When an exponent is 1, the base remains the same.
a^{1} = a
When an exponent is 0, the result of the exponentiation of any base will always be 1, although some
debate surrounds 0^{0} being 1 or undefined. For many applications, defining 0^{0} as 1
is convenient.
a^{0} = 1
Shown below is an example of an argument for a^{0}=1 using one of the previously mentioned
exponent laws.
If a^{n} × a^{m} = a^{(n+m)}
Then a^{n} × a^{0} = a^{(n+0)} = a^{n}
Thus, the only way for a^{n} to remain unchanged by multiplication, and this exponent law
to remain true, is for a^{0} to be 1.
When an exponent is a fraction where the numerator is 1, the n^{th} root of the base is taken.
Shown below is an example with a fractional exponent where the numerator is not 1. It uses both the rule
displayed, as well as the rule for multiplying exponents with like bases discussed above. Note that the
calculator can calculate fractional exponents, but they must be entered into the calculator in decimal
form.
It is also possible to compute exponents with negative bases. They follow much the same rules as
exponents with positive bases. Exponents with negative bases raised to positive integers are equal to
their positive counterparts in magnitude, but vary based on sign. If the exponent is an even, positive
integer, the values will be equal regardless of a positive or negative base. If the exponent is an odd,
positive integer, the result will again have the same magnitude, but will be negative. While the rules
for fractional exponents with negative bases are the same, they involve the use of imaginary numbers
since it is not possible to take any root of a negative number. An example is provided below for
reference, but please note that the calculator provided cannot compute imaginary numbers, and any inputs
that result in an imaginary number will return the result "NAN," signifying "not a number." The
numerical solution is essentially the same as the case with a positive base, except that the number must
be denoted as imaginary.