How to work with negative exponents

How to Work with Negative Exponents

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  • A L G E B R A
  • Table of Contents | Home
  1. 21
  2. Power of a fraction
  3. Subtracting exponents
  4. Negative exponents
  5. Section 2
  6. Exponent 0
  7. Scientific notation
  8. Power of a fraction

“To raise a fraction to a power, raise the numerator and denominator to that power.”

Example 1.

For,  according to the meaning of the exponent, and the rule for multiplying fractions:

How to Work with Negative Exponents

Example 2.   Apply the rules of exponents:   

Solution.   We must take the 4th power of everything.  But to take a power of a power — multiply the exponents:

How to Work with Negative Exponents

Problem 1.   Apply the rules of exponents.

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Do the problem yourself first!

  a)   = x2y2 b)   = 8x327 c)   =
  d)   =
  e)   = x2 − 2x + 1x2 + 2x + 1
  • Perfect Square Trinomial
  • Subtracting exponents
  • In the previous Lesson we saw the following rule for reducing a fraction:
  • “Both the numerator and denominator may be divided by a common factor.”
  • Consider these examples:
2· 2· 2· 2· 2     2· 2 = 2· 2· 2
___2· 2___2· 2· 2· 2· 2 = __1__2· 2· 2

If we write those with exponents, then

22 = 23

In each case, we subtract the exponents.  But when the exponent in the denominator is larger, we write 1-over the difference.

  Example 3.    x3 = x5
x8 =  1 x5

Here is the rule:

How to Work with Negative Exponents

Problem 2.   Simplify the following.  (Do not write a negative exponent.)

  a)   = x3 b)   x2x5 =  1 x3 c)    xx5 =  1x4
  d)   x2x = x e)   = x4 f)   =  1x2

Problem 3.   Simplify each of the following.  Then calculate each number.

  a)   = 23 = 8 b)   2225 =  1 23 = 18 c)    225 =  124 =  1 16
  d)   222 = 2 e)   = −24 = −16.  See Lesson 13.
  f)   =  122 = 14
Example 4.   Simplify by reducing to lowest terms:   

Solution.   Consider each element in turn:

How to Work with Negative Exponents

Problem 4.   Simplify by reducing to lowest terms.  (Do not write negative exponents.

  a)   =   y35x3 b)   b)    = 8a35b3
  c)   =   3z_5x4y3 d)    =   c316
 e)   (x + 1)3 (x − 1)(x − 1)3 (x + 1) = (x + 1)2(x − 1)2

Negative exponents

We are now going to extend the meaning of an exponent to more than just a positive integer.  We will do that in such a way that the usual rules of exponents will hold.  That is, we will want the following rules to hold for any exponents:  positive, negative, 0 — even fractions.

  1. We begin by defining a number with a negative exponent.
  2. It is the reciprocal of that number with a positive exponent.
  3. an is the reciprocal of an.
  Example 5.     2−3 =  123 = 18

The base, 2, does not change.  The negative exponent becomes positive — in the denominator.

Example 6.   Compare the following. That is, evaluate each one:

3−2    −3−2   (−3)−2    (−3)−3

  Answers.  3−2  =   132  =  19 .


−3−2 is the negative of 3−2. (See Lesson 13.)  The base is still 3.

As for (−3)−2, the parentheses indicate that the base is −3:


(−3)−3  =     1   (−3)3  =   1 27 .

A negative exponent, then, does not produce a negative number.  Only a negative base can do that. And then the exponent must be odd

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  Example 7.   Simplify   a2a5 .

Solution.   Since we have invented negative exponents, we can now subtract any exponents as follows:

We now have the following rule for any exponents m, n:

In fact, it was because we wanted that rule to hold that we

We want


Therefore, we define  a−3 as   1 a3 .

a−1 is now a symbol for the reciprocal, or multiplicative inverse, of any number a.  It appears in the following rule (Lesson 5):

Problem 5.   Evaluate the following.

   a)  ( 23 )−1   =   32 . 32  is the reciprocal of  23 .
   b)  ( 23 )−4   =   8116 . 8116  is the reciprocal of the 4th power of   23 .
Problem 6.   Show:  am· bn  =  
ambn .
am· bn  =  am·  1bn   =   ambn .    The definition of division.

Example 9.   Use the rules of exponents to evaluate  (2−3· 104)−2.

Problem 7.   Evaluate the following.

  a)   2−4  =   1 24  =   116   b)   5−2  =   1 52  =   125   c)   10−1  =   1 101  =   110
  d)   (−2)−3  =     1   (−2)3  =    1  −8  =  18
  e)   (−2)−4  =     1   (−2)4  =   1 16 f)   −2−4  =   1 24  =   1 16

g)  (½)−1 = 
2.   2 is the reciprocal of ½.

Problem 8.   Use the rules of exponents to evaluate the following.

   a)   102· 10−4 = 102 − 4 = 10−2 = 1/100.
   b)   (2−3)2   =   2−6   =    1 26   =    1 64
   c)   (3−2· 24)−2   =   34· 2−8   =   3428   =    81 256
   d)   2−2· 2   =   2−2+1   =   2−1   =   12

Problem 9.   Rewrite without a denominator.

  a)   x2x5 = x2−5 = x−3 b)    yy6 = y1−6 = y−5
  c)   = x−3y−4 d)   = a−1b−6c−7
 g)   (x + 1)    x  =  (x + 1)x−1   h)   (x + 2)2(x + 2)6 = (x + 2)−4
  • Example 10.     Rewrite without a denominator, and evaluate:
  • Answer.   The rule for subtracting exponents —
  • — holds even when an exponent is negative.

How to Work with Negative Exponents


= 10−3 + 5 − 2 + 4 = 104 = 10,000.

Exponent 2 goes into the numerator as −2;  exponent −4 goes there as +4.

Problem 10.    Rewrite without a denominator and evaluate.

 a)    222−3 = 22 + 3  = 25 = 32 b)    10210−2 = 102 + 2  = 104 = 10,000
 c)   = 102 − 5 − 4 + 6  = 10−1 =  1 10
 d)   = 25 − 6 + 9 − 7  = 21 = 2
  1. The reciprocal of an.
  2. Reciprocals come in pairs.  The reciprocal of  an  is  an :
  3. And the reciprocal of  an  is  an :
  4. That implies:
  5. Factors may be shifted between the denominator and the numeratorby changing the sign of the exponent.
  Example 11.   Rewrite without a denominator:  
  Answer.    How to Work with Negative Exponents

The exponent 3 goes into the numerator as −3;  the exponent −4 goes there as +4.

Problem 11.    Rewrite with positive exponents only.

 a)    x y−2 = xy2 b)   = c)   =
d)   = e)   =

Problem 12.    Apply the rules of exponents, then rewrite with positve exponents.

a)   = = b)   = =

Section 2: Exponent 0

How to Work with Negative Exponents

Next Lesson:  Multiplying and dividing algebraic fractions

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Working with Negative Exponents

Negative exponentsare a way of writing powers of fractions or decimals without using a fraction or decimal. You use negative exponents as a way to combine expressions with the same base, whether the different factors are in the numerator or denominator. It’s a way to change division problems into multiplication problems.

Example: Instead of writing

How to Work with Negative Exponents

A reciprocal of a number is the multiplicative inverse of the number. The product of a number and its reciprocal is equal to 1.

  • How to Work with Negative Exponents

    How to Work with Negative Exponents

  • The variable x is any real number except 0, and the exponent a is any real number. And, going to the negative side,
  • The following examples show you how to change from positive to negative exponents, and vice versa.

How to Work with Negative Exponents

If you start out with a negative exponent in the denominator, then the negative exponent in the denominator comes up to the numerator with a change in the sign to a positive exponent. For example,

How to Work with Negative Exponents

Then you get,

How to Work with Negative Exponents

Negative Exponents Explained

Mostly everybody knows about exponents and how they work with numbers, right? For example:

How to Work with Negative Exponents

But have you ever thought about having a negative number as a exponent? I’ll show you how it works.

So let’s say we have 2 to the 1st power. That’s 2 Right? Now let’s do 2 to the 2nd power. That equals 4. Keep adding the exponent by 1. You’ll see a pattern

How to Work with Negative Exponents

When we keep adding the exponent by 1, a *2 is added. Now what if we went backwards? 2 two the 1st power is 2 but what is 2 the the 0th power? It is 1. But How? I was wondering why the heck the answer was 1 in algebra class. Its weird. But it’s True. Here’s a photo showing why

How to Work with Negative Exponents

Now what is 2 to the -1 power? we now know that 2 to the 0th power is 1 because 2 to the 1st power is 2 and we divided 2 by 2 to get us 1. So if we divide 1 by 2 it’ll get us 1/2. so 2 to the -1 power is 1/2. we can continue this pattern as shown.

How to Work with Negative Exponents

But If You Don’t want to figure it out that way, there is a easier way by making the exponent positive so it will be easier to solve.

Negative Exponents

Basic RulesSci. Not'nEng. Not'nFractional

Once you've learned about negative numbers, you can also learn about negative powers. A negative exponent just means that the base is on the wrong side of the fraction line, so you need to flip the base to the other side. For instance, “x–2” (pronounced as “ecks to the minus two”) just means “x2, but underneath, as in “.

I know that the negative exponent means that the base, the x, belongs on the other side of the fraction line. But there isn't a fraction line!

How to Work with Negative Exponents

To fix this, I'll first convert the expression into a fraction in the way that any expression can be converted into a fraction: by putting it over “1”. Of course, once I move the base to the other side of the fraction line, there will be nothing left on top. But since anything can also be regarded as being multiplied by 1, I'll leave a 1 on top.

Exponents: Negative exponents – Exponents

A negative exponent helps to show that a base is on the denominator side of the fraction line.  In other words, the negative exponent rule tells us that a number with a negative exponent should be put to the denominator, and vice versa. For example, when you see x^-3, it actually stands for 1/x^3. Not too bad right?


You might be wondering about the fraction line, since there isn't one when we just look at x^-3. However, you can actually convert any expression into a fraction by putting 1 over the number. That's the main reason why we can move the exponents around and solve the questions that are to follow.


Learning this lesson will also help you get one step closer to understanding why any number with a 0 in its exponent equals to 1. There'll be a link to a chart at the end of this lesson that can show you how that relationship comes about. You'll soon understand all the basic properties of exponents!


How to solve for for negative exponents

Let's try working with some negative exponent questions to see how we'll move numbers to the top or bottom of a fraction line in order to make the negative exponents positive. We'll start with regular numbers with a negative exponent, then move on to fractions that have negative exponents on both its numerator and denominator.


Question 1:

Solve 2^-2



As we learned earlier, if we move the number to the denominator, it'll get rid of the negative in the exponent. Then, solving for exponents is easy once we have it in a more calculation-friendly form.


= 1/4


Question 2:

Solve -2^-2



In this case, we've got a negative number with a negative exponent. Again, just move the number to the denominator of a fraction to make the exponent positive. One way you can rewrite the question we're given is the following:


= (-1)(2^-2)


Multiplying in that -1 will turn the equation back into what it was originally. However, keeping the -1 outside helps us work with the negative exponent a little easier and allows us to illustrate what's happening.


  • So moving on from the above, we can continue solving with the negative exponent as we did before.
  • (-1)(2^-2)
  • = (-1)(1/2^2)
  • = (-1)(1/4)
  • = -1/4


As you can see, the final answer we get is negative!.


Fractions with negative exponents


Question 3:




If you ever see a negative exponent on the top of a fraction, you know that if you flip it to the bottom, it'll become positive. The same actually works for negative exponents on the bottom.

If you move it to the numerator, its exponent also becomes positive. With that in mind, let's work through the question. Our first step is just to flip the numerator and denominator to get rid of all the negatives in the exponents.

Then solve as usual with the power rule.  


  1. (3^-2)/(4^-3)
  2. = (4^3)/(3^2)
  3. =64/9


Definitely not as confusing as it first looked, right?  


Here's a good place to take a look at comparing negative and positive exponents and seeing how they behave on a graph.

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