Exponent Calculator — Step-by-Step Solutions

The Exponent Calculator computes the result of raising a base number to any power. It handles positive, negative, zero, and fractional exponents with precision.

Enter a base and an exponent to calculate the result. The tool supports large numbers and displays results in both standard and scientific notation when appropriate.

Base (a)

Exponent (n)

Your calculations will appear here

How to Use the Exponent Calculator

Power Mode: enter a base and exponent to calculate the result. For small integer exponents (up to 10), the calculator shows the full expansion as repeated multiplication.
Scientific Notation Mode: enter any number to convert it into scientific notation (a x 10^n). The step-by-step breakdown shows how the decimal point moves and how the exponent is determined.
Exponent Rules Mode: select a rule (product, quotient, or power), then enter a base and two exponents. The calculator applies the chosen rule and shows the simplified expression alongside the numerical result.
Nth Root Mode: enter the root index and the radicand to compute the nth root. The calculator expresses the root as a fractional exponent and evaluates it step by step.
Negative / Fractional Exponent Mode: enter a base and an exponent as a negative number, a fraction (e.g. 2/3), or both. The calculator breaks down the operation into reciprocals and roots, then computes the final value.

Formula and Theory

An exponent tells you how many times a number (the base) is multiplied by itself. For example, 2^5 means 2 multiplied by itself five times. Exponents are a compact way to represent repeated multiplication and appear throughout algebra, physics, and computer science.

Any non-zero number raised to the power 0 equals 1. By widely accepted convention, 0^0 is also defined as 1 in combinatorics and discrete mathematics. Any number raised to the power 1 equals itself.

A negative exponent indicates a reciprocal: a^(-n) = 1/a^n. A fractional exponent combines powers and roots: a^(m/n) = the nth root of a^m. These two ideas can be combined, so a^(-2/3) means 1 divided by the cube root of a squared.

The three fundamental rules are the product rule (a^m x a^n = a^(m+n)), the quotient rule (a^m / a^n = a^(m-n)), and the power rule ((a^m)^n = a^(m x n)). These rules simplify complex expressions and are essential for solving equations involving exponents.

Worked Examples

Example 1: Basic power calculation

Problem: Calculate 2^8

Solution: 2 multiplied by itself 8 times: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256.

Answer: 256

Example 2: Scientific notation conversion

Problem: Convert 0.00047 to scientific notation

Solution: Move the decimal point 4 places to the right to get 4.7. Since the original number is less than 1, the exponent is negative.

Answer: 4.7 x 10^(-4)

Example 3: Product rule application

Problem: Simplify 5^3 x 5^4

Solution: Using the product rule, 5^3 x 5^4 = 5^(3+4) = 5^7. Evaluating gives 78125.

Answer: 5^7 = 78125

Example 4: Cube root

Problem: Find the cube root of 64

Solution: The cube root of 64 is 4, because 4 x 4 x 4 = 64. Equivalently, 64^(1/3) = 4.

Answer: 4

Example 5: Negative fractional exponent

Problem: Calculate 27^(-2/3)

Solution: 27^(-2/3) means 1 divided by the cube root of 27 squared. The cube root of 27 is 3, and 3 squared is 9, so the result is 1/9.

Answer: 0.111111 (or 1/9)

Frequently Asked Questions

What does 0 raised to the power 0 equal?

By convention, 0^0 is defined as 1 in most mathematical contexts, including combinatorics, set theory, and polynomial evaluation. Some analysis textbooks leave 0^0 undefined, but the convention of 1 is standard for discrete mathematics and computing.

Can you raise a negative number to a fractional exponent?

It depends on the denominator of the fraction. If the root index is odd (e.g. 1/3, 1/5), the result is a real number. If the root index is even (e.g. 1/2, 1/4), there is no real result because you cannot take an even root of a negative number in the real number system.

What is the difference between an exponent and a logarithm?

They are inverse operations. An exponent asks "what is a raised to the power n?", whilst a logarithm asks "to what power must a be raised to get a given value?". For example, 2^3 = 8, and log base 2 of 8 = 3.

Why does any number raised to the power 0 equal 1?

Consider the pattern: a^3 = a x a x a, a^2 = a x a, a^1 = a. Each time the exponent decreases by 1, you divide by a. Following that pattern, a^0 = a^1 / a = 1. This also follows from the quotient rule: a^n / a^n = a^(n-n) = a^0, and any non-zero number divided by itself is 1.

How do I convert between scientific notation and standard form?

To convert to scientific notation, move the decimal point until you have a number between 1 and 10, then count how many places you moved it. That count becomes the exponent of 10 (positive if the original number is large, negative if it is small). To convert back, move the decimal point by the exponent value in the appropriate direction.