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

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

how do we divide exponents by exponents?

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

can we ever go back to just numbers like 2+2

Simple solution: Invent a time machine.

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

Look at the hints. The hints are very helpful for exponents.

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!

when you do the opposite of fractions with a negative exponent, do you always have to do reciprocal;

It's not opposite of fractions actually, it's an equivalent expression. they have the equal sign between them. and their value is the same. You can even use the calculator to make sure of that. 4^-2 = (1/4)^2 they are equal, when, and ONLY when: 1- it's a reciprocal. 2- exponent with the opposite sign. Now let's imagine NOT doing the reciprocal like you said: 4^-2 = 4^2 What's that supposed to be?? Well.. 4^2 _is actually the reciprocal!_ You can make sure of that by multiplying them (use calculator). You should get 1 if they're truly reciprocals. 4^-2 * 4^2 = 1 Yep! So keep in mind that to get the number as it is (not changing its value) you have to BOTH flip the fraction (the reciprocal) and flip the sign of the exponent.

is there any videos on scientific notation in math?

Here's a list of videos with examples of scientific notation: https://www.khanacademy.org/math/pre-algebra/pre-algebra-exponents-radicals#pre-algebra-scientific-notation You can find videos of calculations with scientific notation here: https://www.khanacademy.org/math/pre-algebra/pre-algebra-exponents-radicals#pre-algebra-computing-scientific-notation You can also search for anything you need using the search bar at the top left of the site.

Main content

Algebra basics

Course: Algebra basics > Unit 6

Lesson 2: Negative exponents

Negative exponents review

CCSS.Math:

8.EE.A.1

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, start superscript, minus, n, end superscript, equals, start fraction, 1, divided by, x, start superscript, n, end superscript, end fraction

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

Examples

3, start superscript, minus, 5, end superscript, equals, start fraction, 1, divided by, 3, start superscript, 5, end superscript, end fraction
start fraction, 1, divided by, 2, start superscript, 8, end superscript, end fraction, equals, 2, start superscript, minus, 8, end superscript
y, start superscript, minus, 2, end superscript, equals, start fraction, 1, divided by, y, squared, end fraction
left parenthesis, start fraction, 8, divided by, 6, end fraction, right parenthesis, start superscript, minus, 3, end superscript, equals, left parenthesis, start fraction, 6, divided by, 8, end fraction, right parenthesis, cubed

Practice

Problem 1

Current

Select the equivalent expression.

4, start superscript, minus, 3, end superscript, equals, question mark

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, start superscript, n, end superscript
3	2, cubed, equals, 8
2	2, squared, equals, 4
1	2, start superscript, 1, end superscript, equals, 2
0	2, start superscript, 0, end superscript, equals, 1
minus, 1	2, start superscript, minus, 1, end superscript, equals, start fraction, 1, divided by, 2, end fraction
minus, 2	2, start superscript, minus, 2, end superscript, equals, start fraction, 1, divided by, 4, end fraction

Notice how 2, start superscript, n, end superscript 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 start fraction, x, start superscript, n, end superscript, divided by, x, start superscript, m, end superscript, end fraction, equals, x, start superscript, n, minus, m, end superscript. So...

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

We also know that

\begin{aligned} \dfrac{2^2}{2^3}&=\dfrac{\cancel 2\cdot\cancel 2}{\cancel 2\cdot\cancel 2\cdot 2} \\\\ &=\dfrac12 \end{aligned}

And so we get 2, start superscript, minus, 1, end superscript, equals, start fraction, 1, divided by, 2, end fraction.

Also, recall that x, start superscript, n, end superscript, dot, x, start superscript, m, end superscript, equals, x, start superscript, n, plus, m, end superscript. 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\dfrac{1}{2^2} \\\\ &=\dfrac{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/...”
(13 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...”
  (43 votes)
Mariana Orejel
Posted 3 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...”
(9 votes)
Answer
- Kailyn Bradley
  Posted 3 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...”
  (18 votes)
captain sardine
Posted 3 months ago. Direct link to captain sardine's post “can we ever go back to ju...”
can we ever go back to just numbers like 2+2
Button navigates to signup pageComment on captain sardine's post “can we ever go back to ju...”
(9 votes)
Answer
- SP
  Posted 3 months ago. Direct link to SP's post “Simple solution: Invent a...”
  Simple solution: Invent a time machine.
  Comment on SP's post “Simple solution: Invent a...”
  (11 votes)
Tao
Posted 2 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 2 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...”
  (18 votes)
jaden white
Posted 5 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...”
(9 votes)
Answer
- ncher
  Posted 2 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...”
  (3 votes)
twilliamson554
Posted 3 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...”
(8 votes)
Answer
- Hills01
  Posted a month 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!
  Button navigates to signup page
  (1 vote)
mia.ford22
Posted 5 years ago. Direct link to mia.ford22's post “if a exponent is negative...”
if a exponent is negative what happens to the base
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- ZeroFK
  Posted 5 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)
ajohnson62
Posted 5 years ago. Direct link to ajohnson62's post “when you do the opposite ...”
when you do the opposite of fractions with a negative exponent, do you always have to do reciprocal;
Button navigates to signup pageButton navigates to signup page
(0 votes)
Answer
- Aria
  Posted 5 years ago. Direct link to Aria's post “It's not opposite of frac...”
  It's not opposite of fractions actually, it's an equivalent expression. they have the equal sign between them. and their value is the same. You can even use the calculator to make sure of that.
  
  4^-2 = (1/4)^2
  they are equal, when, and ONLY when:
  1- it's a reciprocal.
  2- exponent with the opposite sign.
  
  Now let's imagine NOT doing the reciprocal like you said:
  4^-2 = 4^2
  
  What's that supposed to be?? Well.. 4^2 is actually the reciprocal!
  You can make sure of that by multiplying them (use calculator). You should get 1 if they're truly reciprocals.
  4^-2 * 4^2 = 1
  Yep!
  
  So keep in mind that to get the number as it is (not changing its value) you have to BOTH flip the fraction (the reciprocal) and flip the sign of the exponent.
  Comment on Aria's post “It's not opposite of frac...”
  (8 votes)
avabeans
Posted 2 years ago. Direct link to avabeans's post “is there any videos on sc...”
is there any videos on scientific notation in math?
Button navigates to signup pageButton navigates to signup page
(0 votes)
Answer
- NatureNotes
  Posted 2 years ago. Direct link to NatureNotes's post “Here's a list of videos w...”
  Here's a list of videos with examples of scientific notation:
  https://www.khanacademy.org/math/pre-algebra/pre-algebra-exponents-radicals#pre-algebra-scientific-notation
  
  You can find videos of calculations with scientific notation here:
  https://www.khanacademy.org/math/pre-algebra/pre-algebra-exponents-radicals#pre-algebra-computing-scientific-notation
  
  You can also search for anything you need using the search bar at the top left of the site.
  Button navigates to signup page
  (7 votes)
2021dylan_carrillo
Posted 4 years ago. Direct link to 2021dylan_carrillo's post “how do i know if the answ...”
how do i know if the answer will be a fraction or not?
Button navigates to signup pageComment on 2021dylan_carrillo's post “how do i know if the answ...”
(2 votes)
Answer
- Elliot Wharton
  Posted 4 years ago. Direct link to Elliot Wharton's post “if it is a fraction you m...”
  if it is a fraction you make it in to the whole number for example 1/5^4 = 5^4 is ^ means exponent
  Comment on Elliot Wharton's post “if it is a fraction you m...”
  (1 vote)