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Negative Exponents

Negative Exponents

Author: Sophia Tutorial
Description:

Simplify an expression with negative exponents.

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Tutorial

what's covered
  1. Negative Exponents

1. Negative Exponents

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:

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 problem a the second way:
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 over 1 equals 1
Our Solution, when we combine the two solutions, we get:
a to the power of 0 equals 1
Our final result

This final result is an important property known as the zero property of exponents:

formula

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. This is illustrated in the following example.

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

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

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 we will also solve this problem another way
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, putting these solutions together gives:
a to the power of short dash 2 end exponent equals 1 over a squared
Our Final Solution

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. Following are the properties of negative exponents.

formula

Properties of Negative Exponents
table attributes columnalign left end attributes row cell a to the power of negative n end exponent equals 1 over n end cell row cell 1 over a to the power of negative n end exponent equals a to the power of n end cell row cell 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 end cell end table
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 exponent moves, the exponent is now positive. This is illustrated in the following 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

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.

Simplifying with negative exponents is much the same as simplifying with positive exponents. It is the advice of the author 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. This is illustrated in the following examples:

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.
Formulas to Know
Properties of Negative Exponents

a to the power of negative n end exponent equals 1 over a to the power of n

1 over a to the power of negative n end exponent equals a to the power of n

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