Exponent Calculator
Calculate any base raised to any power, including fractional exponents.
The base is the number being multiplied.
Use decimals or fractions like 1/2.
Calculated result
32
2 raised to 5
Expression
2^5
Base
2
Number being raised.
Exponent
5
Power applied to the base.
Scientific notation
3.200000e+1
Compact large/small format.
Expanded form
2 × 2 × 2 × 2 × 2
Repeated multiplication.
Exponent Formulas and Rules
Basic power
a^n = a × a × ... × a
Zero exponent
a^0 = 1, when a ≠ 0
Negative exponent
a^-n = 1 ÷ a^n
Fractional exponent
a^(1/n) = ⁿ√a
Power of a power
(a^m)^n = a^(m×n)
Product rule
a^m × a^n = a^(m+n)
Quotient rule
a^m ÷ a^n = a^(m−n)
Power of a product
(ab)^n = a^n × b^n
Variable Explanations
a
Base: the number being raised to a power.
n
Exponent or power applied to the base.
a^n
Base raised to an exponent.
m and n
Exponent values used in exponent rules.
Reciprocal
1 divided by a value.
Root
The inverse operation of a power.
Scientific notation
A compact way to write very large or small numbers.
Exponent Rules Explained
Worked Examples
2^5
Rule: Positive whole exponent
Step: 2 × 2 × 2 × 2 × 2
Answer: 32
5^0
Rule: Zero exponent
Step: Any non-zero base to 0
Answer: 1
10^-3
Rule: Negative exponent
Step: 1 ÷ 10^3
Answer: 0.001
9^(1/2)
Rule: Fractional exponent
Step: √9
Answer: 3
2^3 × 2^4
Rule: Product rule
Step: 2^(3 + 4)
Answer: 128
3^5 ÷ 3^2
Rule: Quotient rule
Step: 3^(5 − 2)
Answer: 27
(4^2)^3
Rule: Power of a power
Step: 4^(2 × 3)
Answer: 4096
10^6
Rule: Scientific notation
Step: 1 × 10^6
Answer: 1,000,000
Negative, Zero, and Fractional Exponents
Zero exponents
In many school contexts, a^0 = 1 when a is not zero.
Negative exponents
A negative exponent creates a reciprocal, not necessarily a negative result.
Fractional exponents
A fractional exponent can represent a root or a root-and-power operation.
Decimal exponents
Decimal exponents are supported when the result is a real number.
0^0
0^0 is undefined or indeterminate depending on context.
Negative bases
Even roots of negative numbers are not real numbers.
Common Exponent Mistakes
Understanding Your Result
Positive whole exponent
Repeated multiplication of the base.
Zero exponent
Equals 1 for non-zero bases.
Negative exponent
Reciprocal of the positive power.
Fractional exponent
Root or root-and-power operation.
Scientific notation
Compact representation of a large or small result.