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Negative Exponents

Author: Sophia
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what's covered
In this lesson, you will learn how to simplify an expression with negative exponents. Specifically, this lesson will cover:

Table of Contents

1. Zero Property

There are a few special exponent properties that deal with exponents that are not positive. The first is considered in the following example, which is worded out 2 different ways:

EXAMPLE

Use the quotient property and solve.

a cubed over a cubed Use the quotient rule to subtract exponents
a to the power of 0 Our Solution

But now we consider the same problem in a second way:

EXAMPLE

Rewrite the exponents as repeated multiplication and solve.

a cubed over a cubed Rewrite exponents as repeated multiplication
fraction numerator a a a over denominator a a a end fraction Reduce out all the a apostrophe s
1 Our Solution

When we combine these two results, we get a to the power of 0 equals 1. This final result is an important property known as the zero property of exponents:

formula to know
Zero Property of Exponents
a to the power of 0 equals 1

Any number or expression raised to the zero power will always be 1.

EXAMPLE

open parentheses 3 x squared close parentheses to the power of 0 Zero power rule
1 Our Solution

2. Negative Exponents

2a. Rule #1

Another property we will consider here deals with negative exponents. Again we will solve the following example in two ways.

EXAMPLE

Use the quotient property and solve.

a cubed over a to the power of 5 Using the quotient rule, subtract exponents
a to the power of short dash 2 end exponent Our Solution

But now we consider the same problem in a second way:

EXAMPLE

Rewrite the exponents as repeated multiplication and solve.

a cubed over a to the power of 5 Rewrite exponents as repeated multiplication
fraction numerator a a a over denominator a a a a a end fraction Reduce three a apostrophe s out of top and bottom
fraction numerator 1 over denominator a a end fraction Simplify to exponents
1 over a squared Our Solution

When we combine these two results, we get a to the power of short dash 2 end exponent equals 1 over a squared. This example illustrates an important property of exponents. Negative exponents yield the reciprocal of the base. Once we take the reciprocal the exponent is now positive. Also, it is important to note a negative exponent does not mean the expression is negative, only that we need the reciprocal of the base. This gives us Rule #1 of the properties of negative exponents.

formula to know
Properties of Negative Exponents
Rule space # 1 colon thin space a to the power of short dash n end exponent equals 1 over n

2b. Rule #2

Negative exponents can be combined in several different ways. As a general rule if we think of our expression as a fraction, negative exponents in the numerator must be moved to the denominator, likewise, negative exponents in the denominator need to be moved to the numerator. When the base with the exponent moves, the exponent is now positive.

EXAMPLE

fraction numerator a cubed b to the power of short dash 2 end exponent c over denominator 2 d to the power of short dash 1 end exponent e to the power of short dash 4 end exponent f squared end fraction Negative exponents on b, d, and e need to flip
fraction numerator a cubed c d e to the power of 4 over denominator 2 b squared f squared end fraction Our Solution

This gives us Rule #2 of the properties of negative exponents.

formula to know
Properties of Negative Exponents
Rule space # 2 colon thin space 1 over a to the power of short dash n end exponent equals a to the power of n

As we simplified our fraction we took special care to move the bases that had a negative exponent, but the expression itself did not become negative because of those exponents. Also, it is important to remember that exponents only affect what they are attached to. The 2 in the denominator of the above example does not have an exponent on it, so it does not move with the d.

2c. Rule #3

What if you had an expression with a fraction and there was a negative exponent applied to the whole fraction?

EXAMPLE

open parentheses a over b close parentheses to the power of short dash 2 end exponent Invert the numerator and denominator and change the negative exponent to a positive exponent.
b squared over a squared Our Solution

The result is simply the reciprocal of the fraction, where each term in the numerator and denominator is raised to a positive power n. This gives us Rule #3 of the properties of negative exponents.

formula to know
Properties of Negative Exponents
Rule space # 3 colon thin space open parentheses a over b close parentheses to the power of short dash n end exponent equals b to the power of n over a to the power of n

3. Simplifying With Negative Exponents

Simplifying with negative exponents is much the same as simplifying with positive exponents. It is advised to keep the negative exponents until the end of the problem and then move them around to their correct location (numerator or denominator). As we do this, it is important to be very careful of rules for adding, subtracting, and multiplying with negatives.

EXAMPLE

fraction numerator x to the power of short dash 5 end exponent x to the power of 7 over denominator x to the power of short dash 4 end exponent end fraction Simplify numerator with product rule, adding exponents
x squared over x to the power of short dash 4 end exponent Use Quotient rule to subtract exponents, be careful with the negatives!
x to the power of 6 Our Solution

EXAMPLE

fraction numerator 4 x to the power of short dash 5 end exponent y to the power of short dash 3 end exponent times 3 x cubed y to the power of negative 2 end exponent over denominator 6 x to the power of short dash 5 end exponent y cubed end fraction Simplify numerator with product rule, adding exponents
fraction numerator 12 x to the power of short dash 2 end exponent y to the power of short dash 5 end exponent over denominator 6 x to the power of short dash 5 end exponent y cubed end fraction Quotient rule to subtract exponents, be careful with negatives!
open parentheses short dash 2 close parentheses minus open parentheses short dash 5 close parentheses equals open parentheses short dash 2 close parentheses plus 5 equals 3
open parentheses short dash 5 close parentheses minus 3 equals open parentheses short dash 5 close parentheses plus open parentheses short dash 3 close parentheses equals short dash 8
2 x cubed y to the power of short dash 8 end exponent Negative exponent needs to move down to denominator
fraction numerator 2 x cubed over denominator y to the power of 8 end fraction Our Solution

EXAMPLE

fraction numerator open parentheses 3 a b cubed close parentheses to the power of short dash 2 end exponent a b to the power of short dash 3 end exponent over denominator 2 a to the power of short dash 4 end exponent b to the power of 0 end fraction In numerator, use power rule with short dash 2, multiplying exponents. In denominator, b to the power of 0 equals 1
fraction numerator 3 to the power of short dash 2 end exponent a to the power of short dash 2 end exponent b to the power of short dash 6 end exponent a b to the power of short dash 3 end exponent over denominator 2 a to the power of short dash 4 end exponent end fraction In numerator, use product rule to add exponents
fraction numerator 3 to the power of short dash 2 end exponent a to the power of short dash 1 end exponent b to the power of short dash 9 end exponent over denominator 2 a to the power of short dash 4 end exponent end fraction Use quotient rule to subtract exponents, be careful with negatives
open parentheses short dash 1 close parentheses minus open parentheses short dash 4 close parentheses equals open parentheses short dash 1 close parentheses plus 4 equals 3
fraction numerator 3 to the power of short dash 2 end exponent a cubed b to the power of short dash 9 end exponent over denominator 2 end fraction Move 3 and b to denominator because of negative exponents
fraction numerator a cubed over denominator 3 squared 2 b to the power of 9 end fraction Evaluate 3 squared 2
fraction numerator a cubed over denominator 18 b to the power of 9 end fraction Our solution

hint
In the previous example it is important to point out that when we simplified 3 to the power of negative 2 end exponent we moved the three to the denominator and the exponent became positive. We did not make the number negative! Negative exponents never make the bases negative, they simply mean we have to take the reciprocal of the base.

summary
You can rewrite any negative exponent as positive using one of these two properties. Any base, b, to a negative exponent, -n can be written as 1 over the same base, b, to a positive exponent, n. The exponent goes from negative to positive. We now have our base and exponent in the denominator of the fraction. It's like we have flipped the fraction.

Similarly, if you have a fraction, 1 over base, b, to a negative exponent, -n, we can write it as the same base b to the positive exponent n. Again, our exponent goes from negative to positive. Instead of the base and exponent being in the denominator of the fraction, we have it written by itself.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

Formulas to Know
Properties of Negative Exponents

Rule space # 1 colon thin space a to the power of negative n end exponent equals 1 over a to the power of n

Rule space # 2 colon space 1 over a to the power of negative n end exponent equals a to the power of n

Rule space # 3 colon space open parentheses a over b close parentheses to the power of negative n end exponent equals b to the power of n over a to the power of n

Zero Property of Exponents

a to the power of 0 equals 1

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