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Division with Exponents

by Ron Kurtus (10 January 2008)

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 are:

• 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. There is a mini-quiz near the end of the lesson.

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*7*7/7*7 =

7*7*7 = 7³

Thus, you can see that 7⁵/7² = 7^5-2 = 7³.

Likewise, x²⁵/x¹⁰ = x^25-10 = x¹⁵.

If you have all variables, x^y/x^z = x^y-z.

Note that the base must be the same to in order to reduce the expression by subtracting exponents. 6⁵ ÷ 2³ and x⁷/y⁴ cannot be reduced.

Dividing by itself

What happens when you divide an exponential number by itself? 11³/11³ = 11^3-3 = 11⁰. The number 11⁰ looks strange, but realizing that a number divided by itself equals 1, you can see that 11⁰ = 1.

Rule: Any number raised to the 0 power equals 1. Thus x⁰ = 7⁰ = 10⁰ = 1. But what about 0⁰? That is a very special case. Although it is not logical, most definitions say that 0⁰ = 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/x^y.

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 divided 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.

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Mini-quiz to check your understanding

1. What is the result of 25³/25^x?

2. What does x⁰/x⁰ equal?

3. What does 1/z^-3 equal?

If you got all three correct, you are on your way to becoming a Champion in Algebra. If you had problems, you had better look over the material again.

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