Explanation of Division with Exponents - Succeed in Understanding Algebra. Also refer to exponential, base, subtract, negative, mathematics, math, maths, Ron Kurtus, School for Champions. Copyright © Restrictions
Division with Exponents
by Ron Kurtus (revised 10 January 2009)
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:
- 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.
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
Thus, you can see that 75/72 = 75−2 = 73.
Variables
Likewise, x25/x10 = x25-10 = x15.
If you have all variables, xy/xz = xy−z.
Must have same base
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 = 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 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/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 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
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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Division with Exponents