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Division with Exponents

by Ron Kurtus (revised 17 August 2012)

When you divide two exponential numbers or variables with the same base, you subtract the exponents. When you divide an exponential number by itself, the exponent is 0. When you divided by a larger exponential with the same base, the exponent is negative.

Questions you may have include:

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.

Numbers

This can be demonstrated with the example of dividing 7*7*7*7*7 by 7*7.

The result is:

(7*7*7*7*7)/(7*7) =

7*7*7 = 73

Since 7*7*7*7*7 = 75 and 7*7 = 72, you get

75/72 = 75−2 = 73

Variables

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 smaler and smaller—apporaching zero—its value remains at 1.

Negative exponents

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.

Summary

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


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