how can you say that 1/1/9 is 9??

1/(1/9). How many 9ths are in one whole? Nine.

how much math is too much math?

any math. all math is to much math

how do we divide exponents by exponents?

you subtract the exponent on the top from the exponent on the bottom.

What happens when zero is put to the zero power, for example 0^0

Interesting question! Consider the following two rules: 1) Any nonzero number to the zero power is 1. 2) Zero to any positive power is 0. If we try to extend both rules to define 0^0, we get different answers. So should 0^0 be 0, 1, or something else? Because of this situation, it is best to call 0^0 indeterminate (though 0^0 is often interpreted as 1). Have a blessed, wonderful day!

Main content

Pre-algebra

Course: Pre-algebra > Unit 11

Lesson 3: Negative exponents

Negative exponents review

Google Classroom

Review the basics of negative exponents and try some practice problems.

Definition for negative exponents

We define a negative power as the multiplicative inverse of the base raised to the positive opposite of the power:

x^{- n} = \frac{1}{x^{n}}

Want to learn more about this definition? Check out this video.

Examples

$3^{- 5} = \frac{1}{3^{5}}$ ‍
$\frac{1}{2^{8}} = 2^{- 8}$ ‍
$y^{- 2} = \frac{1}{y^{2}}$ ‍
${(\frac{8}{6})}^{- 3} = {(\frac{6}{8})}^{3}$ ‍

Practice

Problem 1

Select the equivalent expression.

4^{- 3} = ?

Want to try more problems like these? Check out this exercise.

Some intuition

So why do we define negative exponents this way? Here are a couple of justifications:

Justification #1: Patterns

$n$ ‍	$2^{n}$ ‍
$3$ ‍	$2^{3} = 8$ ‍
$2$ ‍	$2^{2} = 4$ ‍
$1$ ‍	$2^{1} = 2$ ‍
$0$ ‍	$2^{0} = 1$ ‍
$- 1$ ‍	$2^{- 1} = \frac{1}{2}$ ‍
$- 2$ ‍	$2^{- 2} = \frac{1}{4}$ ‍

Notice how

2^{n}

is divided by

2

each time we reduce

n

. This pattern continues even when

n

is zero or negative.

Justification #2: Exponent properties

Recall that

\frac{x^{n}}{x^{m}} = x^{n - m}

. So...

\begin{aligned} \frac{2^{2}}{2^{3}} & = 2^{2 - 3} \\ = 2^{- 1} \end{aligned}

We also know that

\begin{aligned} \frac{2^{2}}{2^{3}} & = \frac{2 \cdot 2}{2 \cdot 2 \cdot 2} \\ = \frac{1}{2} \end{aligned}

And so we get

2^{- 1} = \frac{1}{2}

Also, recall that

x^{n} \cdot x^{m} = x^{n + m}

. So...

\begin{aligned} 2^{2} \cdot 2^{- 2} & = 2^{2 + (- 2)} \\ = 2^{0} \\ = 1 \end{aligned}

And indeed, according to the definition...

\begin{aligned} 2^{2} \cdot 2^{- 2} & = 2^{2} \cdot \frac{1}{2^{2}} \\ = \frac{2^{2}}{2^{2}} \\ = 1 \end{aligned}

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full of brainy brain
Posted 5 years ago. Direct link to full of brainy brain's post “how can you say that 1/1/...”
how can you say that 1/1/9 is 9??
Button navigates to signup pageComment on full of brainy brain's post “how can you say that 1/1/...”
(11 votes)
Answer
- Phillips, Craig
  Posted 3 years ago. Direct link to Phillips, Craig's post “1/(1/9). How many 9ths a...”
  1/(1/9). How many 9ths are in one whole? Nine.
  Comment on Phillips, Craig's post “1/(1/9). How many 9ths a...”
  (56 votes)
Mariana Orejel
Posted 4 years ago. Direct link to Mariana Orejel's post “how do we divide exponent...”
how do we divide exponents by exponents?
Button navigates to signup pageComment on Mariana Orejel's post “how do we divide exponent...”
(7 votes)
Answer
- Kailyn Bradley
  Posted 4 years ago. Direct link to Kailyn Bradley's post “you subtract the exponent...”
  you subtract the exponent on the top from the exponent on the bottom.
  Comment on Kailyn Bradley's post “you subtract the exponent...”
  (25 votes)
Tao
Posted 3 years ago. Direct link to Tao's post “What happens when zero is...”
What happens when zero is put to the zero power, for example 0^0
Button navigates to signup pageComment on Tao's post “What happens when zero is...”
(4 votes)
Answer
- Ian Pulizzotto
  Posted 3 years ago. Direct link to Ian Pulizzotto's post “Interesting question! Co...”
  Interesting question! Consider the following two rules:
  
  1) Any nonzero number to the zero power is 1.
  
  2) Zero to any positive power is 0.
  
  If we try to extend both rules to define 0^0, we get different answers. So should 0^0 be 0, 1, or something else? Because of this situation, it is best to call 0^0 indeterminate (though 0^0 is often interpreted as 1).
  
  Have a blessed, wonderful day!
  Comment on Ian Pulizzotto's post “Interesting question! Co...”
  (23 votes)
LawrenceN
Posted 3 months ago. Direct link to LawrenceN's post “how much math is too much...”
how much math is too much math?
Button navigates to signup pageComment on LawrenceN's post “how much math is too much...”
(10 votes)
Answer
- ANNIIKA :)
  Posted a month ago. Direct link to ANNIIKA :)'s post “any math. all math is to ...”
  any math. all math is to much math
  Comment on ANNIIKA :)'s post “any math. all math is to ...”
  (3 votes)
bentley.pearcy.aka.beyonce
Posted 3 months ago. Direct link to bentley.pearcy.aka.beyonce's post “wowzers i really had a bl...”
wowzers i really had a blast
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(10 votes)
Answer
mia.ford22
Posted 6 years ago. Direct link to mia.ford22's post “if a exponent is negative...”
if a exponent is negative what happens to the base
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(4 votes)
Answer
- ZeroFK
  Posted 6 years ago. Direct link to ZeroFK's post “The base remains the same...”
  The base remains the same. As the page explains, a negative exponent just means "the multiplicative inverse of the base raised to the positive opposite of the power". So a^(-b) = 1/(a^b). The base, a, doesn't change. Only its place in the expression changes.
  Button navigates to signup page
  (8 votes)
jaden white
Posted 6 years ago. Direct link to jaden white's post “what do you do if the que...”
what do you do if the question is not 2 x 2 and its something like 2 x 3 then what number to you put down because if its the same number you can just get rid of one of them but if there not what do you do
Button navigates to signup pageComment on jaden white's post “what do you do if the que...”
(6 votes)
Answer
- ncher
  Posted 3 years ago. Direct link to ncher's post “Look at the hints. The hi...”
  Look at the hints. The hints are very helpful for exponents.
  Comment on ncher's post “Look at the hints. The hi...”
  (1 vote)
twilliamson554
Posted 4 years ago. Direct link to twilliamson554's post “I do not understand why i...”
I do not understand why it becomes a fraction. The intuition does not help either. I know it becomes a fraction, i know the right answer i just do not understand it. For example: 2^-4 i do not get why it becomes 1/2^4
Button navigates to signup pageComment on twilliamson554's post “I do not understand why i...”
(6 votes)
Answer
- Hills01
  Posted 9 months ago. Direct link to Hills01's post “In order to fully underst...”
  In order to fully understand, I found its helpful to draw a number line. On that line would be zero through +8. Then plot or mark where on the line the numbers in the intuitive explanation are. This shows that the neg exponent is doing the opposite of the positive. The number line shows that the positive exponents are high numbers, because the base is increasing exponentially - like multiplication. While the neg exponents are smaller, because the base is exponentially decreasing in size - like division. And it gets more and more small the higher the neg exponent. But it'll never reach or pass zero - which is essentially Zeno's Paradox. I highly recommend checking that out! Understanding Zenos Paradox also helped me understand the depth on neg exponents.
  In math, almost everything has an inverse it seems.
  Pos numbers vs neg numbers,
  Pos exponents vs neg exponents,
  Mult vs division.
  Hope this helps!
  Comment on Hills01's post “In order to fully underst...”
  (1 vote)
zhu31
Posted 4 months ago. Direct link to zhu31's post “This was so helpful”
This was so helpful
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(5 votes)
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CamdynM
Posted 2 months ago. Direct link to CamdynM's post “i need more practice”
i need more practice
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(5 votes)
Answer