Division with Exponents
by Ron Kurtus (revised 14 November 2015)
When you divide two exponential expressions with the same base, you subtract the exponents.
Note: The base of the exponential expression xy is x and the exponent is y.
When you divide an exponential expression by itself, the exponent is 0. When you divided by a larger exponential expression with the same base, the exponent is negative.
Questions you may have include:
- How do you divide numbers raised to a power?
- What does a 0 exponent designate?
- What does a negative exponent designate?
This lesson will answer those questions.
Subtract exponents in division
When you divide exponential numbers or variables with the same base, you subtract the exponents.
This can be demonstrated with the example of dividing 7*7*7*7*7 by 7*7.
The result is:
7*7*7 = 73
Since 7*7*7*7*7 = 75 and 7*7 = 72, you get
75/72 = 75−2 = 73
Likewise, if you divide x25 by x10, you get
x25/x10 = x25-10 = x15
If the variable is raised to some power of another variable, your still subtract the exponents:
xy/xz = xy−z
Must have same base
Note that the base must be the same or a multiple in order to reduce the expression by subtracting exponents.
Neither 55 ÷ 23 nor x7/y4 can be reduced by this method, since the base of each is not the same.
However, in some cases, you can be clever and reduce the numbers to a common base.
Consider 65 ÷ 23. Since 65 = (2*3)5 = 25*35, then
65 ÷ 23 = 25*35/23 = 25 − 3*35 = 22*35
Dividing by itself
What happens when you divide an exponential number by itself?
113/113 = 113−3 = 110
The number 110 looks strange, but realizing that a number divided by itself equals 1, you can see that 110 = 1.
Case of x0
Rule: Any number raised to the 0 power equals 1.
Thus x0 = 70 = 2500 = 1.
Case of 00
But what about 00?
That is a very special case. Although it does not seem logical, most definitions say that 00 = 1.
The way to look at it is by examining fractions to the 0 power.
1/20 = 1/20000 = 1/20000000 = 1/1 = 1
Thus, as the fraction gets smaller and smaller—approaching zero—its value remains at 1.
But what happens when you divide by a number that is larger? If you divide 53 by 57, you will get 53−7 = 5−4.
But also, 5*5*5/5*5*5*5*5 = 1/5*5*5*5 = 1/54. Thus, 5−4 = 1/54.
Likewise, x−3 = 1/x3.
Rule: A negative exponential is the reciprocal of the exponential. x−y = 1/xy.
You subtract the exponents when dividing two exponential numbers or variables with the same base.
When you divide an exponential number by itself, the exponent is 0.
When you divide by a larger exponential with the same base, the exponent is negative.
An exponential number with a negative exponent is the reciprocal of the exponential number.
Increase your understanding by knowing the rules
Resources and references
Exponents: Basic Rules - PurpleMath.com
Exponent Rules - RapidTables.com
Exponents Calculator - CalculatorSoup.com
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