What Are Exponents? - Definition, Properties & Rules

What Are Exponents? - Definition, Properties & Rules
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  • 0:03 What Are Exponents?
  • 2:15 Zero & Negative Exponents
  • 3:39 Product Properties
  • 5:27 Quotient Properties
  • 6:33 Power of a Quotient &…
  • 8:28 Lesson Summary
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Lesson Transcript
Instructor: Kimberly Osborn
Exponents are the mathematical shorthand that tells us to multiply the same number by itself for a specified number of times. This lesson will not only explain how the exponent works but also discuss the seven distinct properties, or rules, that govern its use.

What Are Exponents?

Let's begin this lesson by writing the following on a piece of paper:

3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3

After the fourth or fifth 3, you were probably starting to get bored or annoyed trying to write out the entire expression. Just know, you are definitely not alone in these feelings. Instead of taking the time to write out long strings of numbers like the one above, mathematicians decided to create the exponent. An exponent is just a shorthand way of saying to multiply the same thing by itself for a specified number of times.

Looking at the example, we see that the number 3 is being multiplied by itself 15 times. Using exponents, we can write this one of two ways. The first is using a superscript. You begin by writing the number being multiplied (3) and then the number of times it is being multiplied (15) as a superscript, as shown here.

315

The second method of writing exponents, and the one you will encounter most throughout this lesson, is the use of a caret, or hat. In computer language, this is much easier to create than the superscript. Like with the superscript, you begin by writing the number being multiplied, followed by the caret, and then the number of times it's being multiplied. Using out example, we would write the exponent as 3^15.

Like with many things in math, there is a set of rules, or properties, that govern how we use exponents. In the case of this lesson, we will be discussing seven of these properties as well as exploring examples of how these rules work. Don't worry though, I will include a chart with all of the properties at the end to help you organize everything!

Throughout this lesson you will experience the following important vocabulary:

  • Base is defined as the number being multiplied by itself.
  • Exponent is defined as the number of times you are multiplying the base.

In the example, our base would be the number 3 because it is being multiplied repeatedly, and our exponent would be the number 15 because it describes the number of times 3 is being multiplied by itself.

Zero & Negative Exponents

First, let's look at the zero property of exponents. This is probably one of the easiest properties to remember when it comes to exponents. In simple terms, it means that whenever a number is being multiplied by itself 0 times, the answer will always be 1.

This is written as:

a^0 = 1

However, it is important to note that a cannot equal 0 for this property to work.

'Some examples include:

  • 4^0 = 1
  • 7^0 = 1
  • 2^0 = 1
  • 675^0 = 1

Negative Exponents

Our next property, the negative property of exponents, is also fairly straightforward. It states that whenever you experience a negative exponent, the number you are multiplying becomes a fraction. For example, you might be given the following exponent, 3^(-2). This means that we must rewrite our exponent as a fraction:

1 / 3^2

Once we re-write the exponent, we can then find our solution. In this case, we are multiplying the number 3 by itself two times, so our solution is:

3^(-2) =

1 / (3^2) =

1 / (3 * 3) = 1/9

The property itself is written as:

a^(-b) = 1 / a^b

Again, it is important to note that a cannot equal zero for this property to work.

Product Properties

There are two rules that deal with the product (or multiplication) of two numbers. One is used when the exponents have the same base, while the other is used when the exponents themselves are the same.

Product of Powers

The product of powers property is used when both numbers have the same base but different exponents.

Let's use 2^2 * 2^4 as an example.

In both numbers, we have the same base of 2. The product of powers rule tells us that we can keep this same base and add our exponents together. Thus:

2^2 * 2^4 = 2^(2 + 4)

Now that we have re-written the expression, we can work it out to get a solution:

2^2 * 2^4 =

2^(2 + 4) =

2^(6) = 64

The property itself is written as:

a^b * a^c = a ^ (b + c)

Like with the previous two properties, a cannot equal 0 for this property to work.

Power of a Product

The power of a product property is used when we have different bases but the same exponent.

Let's use 3^2 x 5^2 as an example.

In both numbers, we have the same exponent of 2. The power of product rule tells us that we can keep this same exponent and multiply our two bases together. Thus:

3^2 * 5^2 = (3 * 5)^2

Now that we have re-written the expression, we can work it out to get a solution of:

3^2 * 5^2 =

(3 * 5)^2 =

(15)^2 = 225

The property itself is written as:

b^a x c^a = (b x c)^a

In this case, b and c cannot equal 0 for this property to work.

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