# Negative Exponents

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**Negative exponents are exponents that have a negative value. They indicate that the base of a number should be inverted or taken to the reciprocal.**

For example, the expression x^(-2) is the same as 1/x^2 or the reciprocal of x squared. Negative exponents can represent very small or very large numbers, typically by multiplying a coefficient by 10 raised to a negative power.

In this article, we will cover:

- Negative Exponents Rules
- What is the Difference Between Negative Exponents and Positive Exponents?

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Negative exponents also use in more abstract mathematical contexts, like complex numbers and polynomials. Also, it can be used for solving equations that involve physical units and other advanced mathematical concepts.

It’s important to understand the properties and rules for working with negative exponents to evaluate and simplify expressions that involve them properly.

## Negative Exponents Rules

There are a few rules for exponents that are only for negative exponents. Here are some rules working with negative exponents that you should be familiar with:

### 1. a^(-n) = 1/a^n

This rule states that when you have a negative exponent, you can simplify the expression to get the solution by taking the reciprocal of the base raised to the positive exponent.

### 2. (a^n)^(-m) = a^(-nm)

This rule states that when you have a term raised to a power, and the power itself is raised to another power, you can simplify the expression by dividing the exponent by the power.

### 3. (ab)^n = a^n * b^n

This rule states that when you have a product of two or more bases raised to a power, you can simplify the expression by raising each base to the power individually.

### 4. (a/b)^n = a^n / b^n

This rule states that when you have a quotient of two bases raised to a power, you can simplify the expression by raising each base to the power individually.

### 5. a^n * a^m = a^(n+m)

This rule states that when you have the product of two or more powers of the same base, you can simplify the expression by adding the exponents.

It’s important to remember these rules and practice applying them so that you can easily simplify and evaluate expressions that involve negative exponents. Also, practicing these rules by attempting worksheets or sample questions can help you cement your understanding of the concept well, whether it is the multiplication of negative exponent or any other application.

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## What is the difference between negative and positive exponents?

The main difference between negative and positive exponents is how they affect a number’s base.

A positive exponent indicates that a number is multiplied by itself several times. Let’s see an example step by step, 2^3 = 2*2*2 = 8, which means 2 is multiplied by itself 3 times.

On the other hand, a negative exponent indicates that the reciprocal of a number is being multiplied by itself a certain number of times. The reciprocal of a number is the number flipped upside down, 1/number. For example, 2^(-3) = 2*2*2 = 1/8, which means the reciprocal of 2 is multiplied by itself 3 times.

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Another way to understand negative exponents is that it’s the same as taking the reciprocal of the number with the positive exponent. So, for example 2^3 = 8 and 2^(-3) = 1/8. They both are reciprocal to each other.

So, in simple terms, positive exponents are used to represent a number being multiplied by itself, while negative exponents are used to representing the reciprocal of that same number being multiplied by itself.

**Gloria Mathew** writes on math topics for K-12. A trained writer and communicator, she makes math accessible and understandable to students at all levels. Her ability to explain complex math concepts with easy to understand examples helps students master math. LinkedIn

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