I went scrounging for some information about why you know to put a negative exponent in a denominator. I found a very good explanation for why we know it's true because it keeps the rules for adding exponents in place. However, I came across another explanation for why it really represents a reciprocal and for some reason, I'm not following it. Here's why: 2/3 is the reciprocal of 3/2 because when you multiply the two, you get 1. However, how could 3^-2/1 be the reciprocal of 1/3^2? It seems to me, for it to really be a reciprocal, it would have to be 1/3^-2. How else would the multiplication work out to be 1?
How are negative exponents reciprocals?
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mmm4444bot
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The reciprocal of 1/3^2 is 9.
3^(-2) is not 9.
3^2 is 9.
The English in your post is so convoluted, I cannot resolve it.
Are you sure that you understand the definition of the word "reciprocal" ? :wink:
Sorry...
mmm444bot, I think maybe the problem is that what I read on a website about 3^-2 just may not be true so it's hard to understand what I'm trying to describe. The website claimed that 1/3^2 is the reciprocal of 3^-2/1. I didn't see how that could be true. I didn't see how you could get the number 1 by multiplying 1/3^2 by 3^-2/1. And, thanks, Jeff.
mmm444bot, I think maybe the problem is that what I read on a website about 3^-2 just may not be true so it's hard to understand what I'm trying to describe. The website claimed that 1/3^2 is the reciprocal of 3^-2/1. I didn't see how that could be true. I didn't see how you could get the number 1 by multiplying 1/3^2 by 3^-2/1. And, thanks, Jeff.
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pka
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Well you don't get 1!
\(\displaystyle \dfrac{1}{3^2}\dfrac{3^{-2}}{1}=\dfrac{1}{3^4}\)
This is true \(\displaystyle \dfrac{1}{3^2}\dfrac{3^{2}}{1}=1\)
HallsofIvy
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If you could tell us what the web site is, I would like to check that to see if there is an error or you are misunderstanding. \(\displaystyle 3^{-2}= \frac{1}{3^2}\), not \(\displaystyle 1/3^{-2}\). And \(\displaystyle \frac{1}{3^2}\) is the reciprocal of \(\displaystyle 3^2\) not \(\displaystyle 3^{-2}\).
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this is the website...
http://www.themathpage.com/alg/negative-exponents.htm
this is the quote "It is the reciprocal of that number with a positive exponent."
Either they've misused the word reciprocal here, or I've misunderstood what they're trying to say.
http://www.themathpage.com/alg/negative-exponents.htm
this is the quote "It is the reciprocal of that number with a positive exponent."
Either they've misused the word reciprocal here, or I've misunderstood what they're trying to say.
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| -------------------------------------------- |
We begin by defining > > a number with a negative exponent. < <
| a−n | = | 1 an |
** > > It < < is the reciprocal of that number with a positive exponent.
a−n is the reciprocal of an.
As I read this, the pronoun, "It," stands for the phrase "a number with a negative exponent."
Then ** could be rewritten as:
"A number with a negative exponent is the reciprocal of that number with a positive exponent."
As long as a is not zero, then this is true. Look at the equation in the triple box above line ** .
pka
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I see nothing that is not standard and correct on that webpage.
\(\displaystyle a^{-n}\) is the reciprocal of \(\displaystyle a^n\). That is what it says.