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Any time you work with expressions that contain exponents, you have to follow a specific set of rules that are not the same as when you are working with expressions that do not contain exponents. This lesson will describe the rules associated with dividing terms that contain exponents.

So, What Is an Exponent Anyway?

An exponent is a number written as a superscript to another number. It looks like this: 23 or 2^3The exponent of a number tells you how many times to multiply that number to itself. So, in the earlier example, 2^3 means 2*2*2, which is equal to 8.

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Using the caret or ^ symbol is another way to write an exponent that can be easier when typing.Exponents make large multiplication problems easier to write. So, 7*7*7*7*7 = 75, because the equation is asking you to multiply 7 to itself 5 times.

Division of Exponents Means Subtraction

There are certain laws that govern working with exponents. The rule dealing with dividing expressions containing exponents is what this lesson is all about.When you have two exponential expressions that have the same base, you can easily divide one from another. All you have to do in this instance is subtract the exponent of the denominator (the bottom number of a fraction) from the exponent of the numerator (the top number of a fraction).

Here’s an example: 57 / 52. To simplify this expression, just subtract the exponents: 7 – 2 = 5. So, the answer is 55.

Let’s look at how this works. If we write out the multiplication of each exponent we get:(5*5*5*5*5*5*5) / (5*5)Two of the fives in the numerator will cancel out with the two fives in the denominator, which leaves us with:5*5*5*5*5To write this with exponents, we just count up the number of fives – there happen to be 5 of them – so the simplified answer is 55.This simplification works with all exponential expressions where the base is the same for each term. If the base is different, no simplification can be done.Let’s try another example: Simplify y8 / y6.

Just like before, to simplify this expression, just subtract the exponents: 8 – 6 = 2, so y8 / y6 = y2.The rule also applies if one or more of the exponents are negative. Simplify b-2 / b6.

Again, just subtract the exponents, making sure to subtract them in the proper order. -2 – 6 = -8. So, b-2*b6 = b-8.

What if the Terms Have Different Bases?

If the terms you are working with have different bases, there is not much you can do to simplify the expression.For example: Simplify c12 / 47. Since the bases for each term (c and 4) are different, nothing can be done to simplify this expression, and you are left with c12 / 47.Simplify x5 / 23.

The same rule applies to this example. Because the bases are not the same, nothing can be done to simplify the expression. The answer is x5 / 23.

The only exception to this rule is if both the bases are numbers. Then, to simplify, you can simplify each term and then divide.Simplify 43 / 23. 43 = 64 and 23 = 8. You can simplify further and then divide.

64/8 = 8.

Lesson Summary

Exponents are numbers written as superscripts that tell you how many times to multiply the base number to itself. In order to divide one exponential expression by another with the same base, you just need to subtract the exponents. This works if the exponents are positive or negative, but only if the bases are the same. If the terms have different bases, there is not much that can be done to simplify the expression.

Learning Outcomes

Because of the knowledge you take from this video lesson, you could realize these objectives:

  • Provide the meaning and function of an exponent
  • Divide one exponential expression by another when the bases are the same
  • Recall whether you can simplify exponential expressions when the bases are not the same

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