Exponent calculator
Powers, roots of any degree and logarithms in any base, including correct real answers for odd roots of negative numbers.
How it works
An exponent, or power, indicates how many times a base number is multiplied by itself, written x to the power of n. Related operations include roots, the nth root of x equals x to the power of 1 over n, and logarithms, the inverse of exponentiation.
- Choose a section
Power, root, or logarithm, each computed independently.
- Enter the numbers
Results update as you type, with a reason shown when a result is not a real number.
- Read the result
Each section shows its own answer.
Exponent, root and logarithm rules
Power rules
x^a × x^b = x^(a+b); x^a / x^b = x^(a-b); (x^a)^b = x^(ab); x^0 = 1 for any non-zero x; x^(-n) = 1/x^n.
Root rules
The nth root of x equals x^(1/n). The nth root of (xy) equals the nth root of x times the nth root of y. An odd-degree root of a negative number is a real, negative number: the cube root of -8 is -2, because -2 × -2 × -2 = -8. An even-degree root of a negative number, such as a square root, is not a real number.
Logarithm rules
log(xy) = log(x) + log(y); log(x/y) = log(x) - log(y); log(x^n) = n × log(x); log base b of x = ln(x) / ln(b).
What is and is not a real number
0 to the power of 0 is treated as 1 by convention, though it is mathematically indeterminate. A negative base raised to a non-whole exponent, and any logarithm of a number that is 0 or negative, are not real numbers; this calculator explains why instead of showing an incorrect value.
Frequently asked questions
What are the basic rules of exponents?
Key rules: x^a times x^b equals x^(a+b); x^a divided by x^b equals x^(a-b); (x^a)^b equals x^(ab); x^0 equals 1 for any non-zero x; x^(-n) equals 1 divided by x^n. For example, 2^3 times 2^4 equals 2^7, which is 128, and 2^(-3) equals 1/8. These rules let you simplify complex exponential expressions.
How do you calculate the root of a negative number?
An odd-degree root of a negative number is a real number: the cube root of -8 is -2, since -2 times -2 times -2 equals -8, and the 5th root of -32 is -2. An even-degree root of a negative number, such as a square root, is not a real number, because no real number multiplied by itself an even number of times gives a negative result.
What is the difference between log and ln?
Log, the common logarithm, traditionally uses base 10: log base 10 of 1000 equals 3, because 10^3 equals 1000. Ln, the natural logarithm, uses base e, approximately 2.71828: ln(e) equals 1. The relationship is log base b of x equals ln(x) divided by ln(b). The calculator shows both for whatever number you enter, plus any custom base.
What is 0 raised to the power of 0?
0^0 is mathematically indeterminate; different branches of mathematics use different conventions. In combinatorics and most calculators, including this one, it is defined as 1 so that formulas keep working smoothly. There is no single universal answer, so treat this as a convention rather than a proven fact.
Why does the logarithm of a negative number or zero show no result?
The logarithm of a number that is zero or negative is undefined in the real number system, because no real exponent of a positive base produces a result that is zero or negative. The calculator shows a plain explanation instead of a wrong number. Logarithms of negative numbers require complex numbers, which are outside this calculator's scope.