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Exponential Properties Involving Quotients

Subtract exponents to divide exponents by other exponents

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Exponential Properties Involving Quotients

Exponential Properties Involving Quotients 

The rules for simplifying quotients of exponents are a lot like the rules for simplifying products.

Let’s look at what happens when we divide x7 by x4:


You can see that when we divide two powers of x, the number of x’s in the solution is the number of x’s in the top of the fraction minus the number of x’s in the bottom. In other words, when dividing expressions with the same base, we keep the same base and simply subtract the exponent in the denominator from the exponent in the numerator.


Quotient Rule for Exponents: xnxm=x(nm)

When we have expressions with more than one base, we apply the quotient rule separately for each base:

Now let’s see what happens if the exponent in the denominator is bigger than the exponent in the numerator. For example, what happens when we apply the quotient rule to x4x7?

The quotient rule tells us to subtract the exponents. 4 minus 7 is -3, so our answer is x3. A negative exponent! What does that mean?




Well, let’s look at what we get when we do the division longhand by writing each term in factored form:


Even when the exponent in the denominator is bigger than the exponent in the numerator, we can still subtract the powers. The x’s that are left over after the others have been canceled out just end up in the denominator instead of the numerator. Just as x7x4 would be equal to x31 (or simply x3), x4x7 is equal to 1x3. And you can also see that 1x3 is equal to x3. We’ll learn more about negative exponents shortly.

Simplifying Expressions 

Simplify the following expressions, leaving all exponents positive.


a) x2x6

Subtract the exponent in the numerator from the exponent in the denominator and leave the x’s in the denominator: x2x6=1x62=1x4

b) a2b6a5b

Apply the rule to each variable separately: a2b6a5b=1a52b611=b5a3



Simplify each of the following expressions using the quotient rule.

Example 1



Example 2



Example 3


c) a5b4a3b2=a53b42=a2b2


Evaluate the following expressions.

  1. 5652
  2. 6763
  3. 34310
  4. 223252
  5. 335237

Simplify the following expressions.

  1. a3a2
  2. x5x9
  3. x6y2x2y5
  4. 6a32a2
  5. 15x55x
  6. 25yx620y5x2

Review (Answers)

To view the Review answers, open this PDF file and look for section 8.3. 

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When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression 32^4, 32 is the base, and 4 is the exponent.


Exponents are used to describe the number of times that a term is multiplied by itself.


The "power" refers to the value of the exponent. For example, 3^4 is "three to the fourth power".

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