# 8 Exponent Rules You Need to Know for Your Next Math Test

When you're dealing with exponents, numbers can get very big (or very small) very quickly. Therefore, it's helpful to have some short cuts.

When you've familiarized yourself with the **exponent rules**, also called the laws of exponents, you can apply them to mathematical problems that would otherwise take more effort and time.

## What Is an Exponent?

An exponent can be defined as a base value *a* that is raised to the power *n*, which would look like *a*^{n}. In ordinary language, this is really just a way of saying it's a number that is multiplied by itself a certain amount of times. Like many concepts in mathematics, exponents are a symbolic notation that create a simplified form of writing an equation.

For instance, if you have 2 raised to the power of 3, (which is also known as 2 cubed; the exponent three is "cubed," and exponent two is "squared"), you must perform the following equation:

2

^{3}= 2 x 2 x 2 = 8

Two is the base number (or base value), and three is the exponent (or exponent value). Base values are the big number in standard font size, exponent values are the number written in the superscript.

Therefore, 2 to the power of 3 — also known as "two the third power" — is 8.

## Why Are Exponent Rules Helpful?

As you can imagine, especially if you have any familiarity with mathematics, things can get complicated quickly. If you raised a number to the power of eight, for example, you'd quickly be getting to some pretty big numbers. And sometimes there will be negative numbers raised to a power, and numbers raised to negative exponents, fractional exponents, and so on.

That's why exponent rules are so handy. If you know them, or keep a handy exponent rules chart, you will have a shorthand for performing the necessary calculations to get your answer. By knowing the right exponent rule (also known as the power rule) for the equation, you can save time and know you're on the way to a correct answer.

## 8 Essential Exponent Rules With Examples

Here are eight exponent rules to master. If you have a chart, or if you want to make a chart, you can have a reference more or less at your fingertips.

### The Product Rule

This rule states that if you need to multiply two exponential expressions with the same base, you can add the exponents together and then raise the base to the sum of the exponents. The product law of exponents can be written symbolically as follows: *a*^{n}* x a*^{m}* = a*^{m+n}.

**Example**:

3

^{3}x 3^{4}= 3^{3+4}

3

^{3+4}= 3^{7}

3

^{7}= 2,187

### The Quotient Rule

Also known as the quotient law, this rule states that if you have two expressions with the same base, you can subtract the exponents and then raise the base to that power. The quotient rule of exponents can be written symbolically as follows: *(a*^{n}*)/(a*^{m}*) = a*^{n–m}.

**Example**:

(4

^{6})/(4^{3}) = 4^{6–3}

4

^{6–3}= 4^{3}

4

^{3}= 64

### The Zero Exponent Rule

Also known as the zero power rule, this expression says that when a number is raised to the power of zero, the answer will be 1. The only exception is raising zero to the zero power, which is an indeterminate expression. The zero exponent law can be written symbolically as follows: *a** = 1*.

**Example**:

45 = 1

### The Identity Exponent Rule

The identity law says that any number raised to the power of 1 is that number itself (the same value). The identity exponent law can be written symbolically as follows: *a*^{1}* = a*.

**Example**:

7

^{1}= 7

### The Negative Exponent Rule

This law says that any number raised to a number of negative value should be solved using the reciprocal. Therefore, the base number and the exponent is placed in the denominator, with a one above it, and the sign of the exponent is changed to a positive exponent.

The negative exponent law can be written symbolically as follows: *a*^{-n}* = 1/(a*^{n}*)*.

**Example**:

5

^{-3}= 1/(5^{3})

1/(5

^{3}) = 1/125

1/125 = 0.008

### The Power of a Power Rule

The power of a power law can be written symbolically as follows: (a^n)^m = a^nm.

**Example**:

(4

^{2})^{6}= 4^{2x6}

4

^{2x6}= 4^{12}

4

^{12}= 16,777,216

### The Power of a Product Rule

When you have an expression where a product is raised to an exponent, distribute the exponent to each number in the product (the technical term for these is "multiplicand"). The power of a product law can be written symbolically as follows: *(ab)*^{n}* = a*^{n}* x b*^{n}.

**Example**:

(2x9)

^{4}= 2^{4}x 9^{4}

2

^{4}x 9^{4}= 16 x 6,561

16 x 6,561 = 104,976

### The Power of a Quotient Rule

This law of exponents applies when you have an expression written as a fraction or a quotient, that is raised to a power. The law states that this is equal to both the numerator and the denominator being raised to that power.

The power of a quotient law can be written symbolically as follows: *(a/b)*^{n}* = a*^{n} */ b*^{n}.

**Example**:

(3/5)

^{2}= 3^{2}/5^{2}

3

^{2}/5^{2}= 9/25

9/25 = 0.36

Now That's Radical

These aren't all the laws of exponents that you might come across, but they are the most commonly taught. Different rules, such as the fractional exponent law, involve the use of other mathematical symbols such as radicals.

Original article: 8 Exponent Rules You Need to Know for Your Next Math Test

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