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High School Algebra I: Help and Review25 chapters | 292 lessons

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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.

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:

- Positive exponents
- Negative exponents
- Zero exponents
- Rational exponents

**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** 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

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

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 = âˆš*x*

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

8^2/3 = 3âˆš(8^2) = 3âˆš64 = 4

*m*^5/2 = âˆš(*m*^5)

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

- When multiplying numbers that have exponents, you add the exponents: (
*a*^2)(*a*^5) =*a*^(2+5) =*a*^7 - When dividing numbers that contain exponents, you subtract the exponents:
*y*^5/*y*^3 =*y*^(5-3) =*y*^2 - When raising a number with an exponent to another power, you multiply the exponents: (
*m*^3)^4 =*m*^(3 * 4) =*m*^12

For all these rules, the base number (or variable) must be the same. You cannot combine expressions with different bases.

*x*^2 + *n*^6 cannot be simplified. Neither can 4^2/*m*^8.

**Exponents** are superscript numbers that tell you how many times to multiply a number by itself. That number, or variable, is called the **base**. There are certain rules that help you to work with exponents. These rules only work if the bases that you are working with are the same.

There are 4 kinds of exponents:

- Positive exponents (deal with positive numbers)
- Negative exponents (deal with negative numbers)
- Zero exponents (an expression with 0 as the exponent and is equal to 1)
- Rational exponents (exponents that are fractions)

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High School Algebra I: Help and Review25 chapters | 292 lessons

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