What is an exponent?

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. Likewise, 4² = 4 × 4, etc. If the exponent is 3, in the example 5³, then the result is 5 × 5 × 5.

If you wish to do exponentiation by hand, do the following:

Determine the base and the power it's raised to, for example, 3⁵.
Write the base the same number of times as the exponent. 3 3 3 3 3

Place a multiplication symbol between each base. 3 × 3 × 3 × 3 × 3.

Multiply! 3 × 3 × 3 × 3 × 3 = 243.

It's easy with small numbers, but for bases that are large numbers, decimals, or when they are raised to a power that's very large or negative, don't hesitate to use our tool.

Negative exponent calculator

The concept is rather simple when the exponent is positive, but what happens when the exponent is negative? By the definition, if it is -2, we would multiply the base times itself negative two times. In actuality, what is happening here, we take the reciprocal of the base and change the negative exponent to positive and proceed as usual. If you'd like to work it out by hand, do the following:

Determine the base and the exponent.
Write the reciprocal of the base and change the sign of the exponent to positive
Write the reciprocal of the base the same number of times as the exponent.
Place a multiplication symbol between each.
Multiply and get the result.

Here's a quick example: 5⁻⁴ = (1/5)⁴ = (1/5) × (1/5) × (1/5) × (1/5) = 1/625 = 0.0016

Example

Exponent Calculator

An exponent represents how many times a number, called the base, is multiplied by itself. The goal of using an exponent calculator is to quickly calculate the value of a number raised to a power.

The general approach to calculating exponents includes:

Identifying the base and the exponent (power).
Raising the base to the given power (multiplying the base by itself the number of times specified by the exponent).
Applying mathematical operations if the exponent is negative, fractional, or zero.

Calculating Positive Exponents

A positive exponent indicates how many times to multiply the base by itself. The general form of a positive exponent is:

\[ a^n \]

Example:

If the expression is \( 2^3 \), the solution is:

Step 1: Multiply the base (2) by itself three times: \( 2 \times 2 \times 2 = 8 \).

Calculating Negative Exponents

A negative exponent represents the reciprocal of the base raised to the positive exponent. The general form of a negative exponent is:

\[ a^{-n} = \frac{1}{a^n} \]

Example:

If the expression is \( 2^{-3} \), the solution is:

Step 1: Convert to a positive exponent: \( 2^{-3} = \frac{1}{2^3} \).
Step 2: Calculate the positive exponent: \( \frac{1}{2^3} = \frac{1}{8} \).

Calculating Fractional Exponents

A fractional exponent represents a root of the base raised to the numerator of the fraction. The general form of a fractional exponent is:

\[ a^{\frac{m}{n}} = \sqrt[n]{a^m} \]

Example:

If the expression is \( 8^{\frac{1}{3}} \), the solution is:

Step 1: Calculate the cube root of 8: \( 8^{\frac{1}{3}} = 2 \).

Real-life Applications of Exponents

Exponents are used in various fields, including:

Calculating compound interest in finance (e.g., determining the growth of an investment over time).
Describing exponential growth or decay (e.g., population growth, radioactive decay).
Working with scientific notation to express very large or very small numbers (e.g., \( 1.23 \times 10^6 \)).

Common Operations with Exponents

Multiplying Exponents with the Same Base: \( a^m \times a^n = a^{m+n} \)

Dividing Exponents with the Same Base: \( \frac{a^m}{a^n} = a^{m-n} \)

Raising an Exponent to Another Power: \( (a^m)^n = a^{m \times n} \)

Zero Exponent: \( a^0 = 1 \) (for any nonzero base).

Exponent Calculator Examples Table
Exponent Type	Description	Steps to Calculate	Example
Positive Exponent	Calculating a base raised to a positive power.	Multiply the base by itself as many times as indicated by the exponent.	For the expression \( 2^3 \), multiply \( 2 \times 2 \times 2 = 8 \).
Negative Exponent	Calculating a base raised to a negative power.	Convert the negative exponent to a reciprocal (invert the base). Calculate the positive exponent and find the reciprocal.	For the expression \( 2^{-3} \), convert to \( \frac{1}{2^3} = \frac{1}{8} \).
Fractional Exponent	Calculating a base raised to a fractional power.	Find the root indicated by the denominator of the fraction. Raise the base to the numerator of the fraction.	For the expression \( 8^{\frac{1}{3}} \), calculate the cube root of 8 to get \( 2 \).
Real-life Applications	Applying exponents to solve practical problems.	To calculate compound interest over time. To describe exponential growth, such as population or bacteria growth.	If an investment grows by 5% annually, the value after 3 years is \( P(1.05)^3 \), where \( P \) is the principal amount.