Exponent Calculator

The exponent calculator will calculate the value of any base raised to any power. An exponent is a way to represent how many times a number, known as the base, is multiplied by itself. It is represented as a small number in the upper right hand corner of the base. For example: x² means you multiply x by itself two times, which is x * x. If the exponent is 3, in the example 5³, then the result is 5 * 5 * 5.

Exponent Definition

An exponent is the small number located in the upper, right-hand position of an exponential expression (baseexponent), which indicates the power to which the base of the expression is raised.

The following exponential expression shows how the exponent is used in raising the base:

4³ = 4 x 4 x 4 = 64

4 x 4 = 16

4 x 16 = 64

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

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

The exponent formula is:

a ⁿ = a×a×...×a

n times

The base a is raised to the power of n, is equal to n times multiplication of a.

For example:

2⁵ = 2×2×2×2×2 = 32

Multiplying exponents

aⁿ ⋅ a^m = a^n+m

Example: 2³ ⋅ 2⁴ = 2⁽³⁺⁴⁾ = 2⁷ = 128

aⁿ ⋅ bⁿ = (a ⋅ b) ⁿ

Example: 3² ⋅ 4² = (3⋅4)² = 12² = 144

Dividing exponents

aⁿ / a^m = a^n-m

Example: 2⁵ / 2³ = 2^(5-3) = 2² = 4

aⁿ / bⁿ = (a / b)ⁿ

Example: 8² / 2² = (8/2)² = 4² = 16

Power of exponent

(aⁿ)^m = a^n⋅m

Example: (2³)⁴ = 2^{(3 ⋅ 4)} = 2¹² = 4096

Radical of exponent

^m√(aⁿ) = a^n/m

Example: ²√(2⁶) = 2^{(6 / 2)} = 23 = 8

Negative exponent

a ^-n = 1 / a ⁿ

Example: 2^-3 = 1 / 2³ = 1 / 8 = 0.125

Zero exponent

a ⁰ = 1

Example: 4⁰ = 1

below is an another example of an argument for a⁰=1 using one of the previously mentioned exponent laws.

If aⁿ × a^m = 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.

When an exponent is a fraction where the numerator is 1, the n^th root of the base is taken.

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.