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Exponent: Definition & Properties

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  • 0:02 Definition of an Exponent
  • 0:43 Positive Exponents & Examples
  • 1:14 Negative Exponents & Examples
  • 2:30 Zero / Rational…
  • 3:31 Equations with…
  • 4:25 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe
An exponent tells you how many times to use a number in a multiplication problem. This lesson will define the properties of exponents and how to interpret them. There will also be a quiz at the end of the lesson.

Definition of an Exponent

An exponent is a number that indicates how many times you should multiply a number to itself. For example, 4^2 means multiply 4 by itself 2 times, or 4 * 4 = 16. Therefore, 4^2 = 16.

The exponent is written as a superscript number after the number being multiplied, which is called the base. In the example we just looked at, the number 2 is the exponent and the 4 is the base.

There are 4 types of exponents:

  1. Positive exponents
  2. Negative exponents
  3. Zero exponents
  4. Rational exponents

Positive Exponents & Examples

Positive exponents are exponents that are positive numbers. There is no special trick to working with positive exponents, just multiply the base to itself the number of times indicated by the exponent.

Here are a couple of examples of positive exponents:

3^5 = 3 * 3 * 3 * 3 * 3 = 243

7^3 = 7 * 7 * 7 = 343

Negative Exponents & Examples

Negative exponents are negative numbers that are being used as exponents. For example, 2^-4.

A negative exponent is simplified by placing the base (with the exponent) in the denominator of a fraction with 1 as the numerator.

2^-4 = 1 / (2^4) = 1/16

Here is how that works:

  • 2^4 = 16
  • 2^3 = 8
  • 2^2 = 4
  • 2^1 = 2

For each step as the exponent is decreased, the solution is divided by 2. The pattern continues as you keep decreasing the exponent.

  • 2^0 = 1
  • 2^-1 = 1 / 2
  • 2^-2 = 1 / 4 (1 / 2^2)
  • 2^-3 = 1 / 8 (1 / 2^3)

This rule applies to all negative exponents. Here's some more examples:

3^-4 = 1 / 3^4 = 1/81

x^-7 = 1 / x^7

Zero Exponents & Examples

An expression with 0 as the exponent is equal to 1. It does not matter what the base is, if the exponent is 0 the simplification is 1. Now, let's look at some specific examples:

25^0 = 1

b^0 = 1

Rational Exponents & Examples

A rational exponent is an exponent that is a fraction: for example, x^1/2. The 1/2 is a rational exponent. Rational exponents are a different way to write a radical expression where the top number of the fraction is the power and the bottom number is the root.

So, x^1/2 = √x

Now let's look at some examples of rational exponents:

8^2/3 = 3√(8^2) = 3√64 = 4

m^5/2 = √(m^5)

Equations with Exponents & Examples

There are specific rules to help you simplify problems that include exponents.

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