How Do Negative Exponents Work in Math?

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Discussion Overview

The discussion revolves around the concept of negative exponents in mathematics, specifically how to manipulate expressions involving them. Participants explore different methods for handling negative exponents, including their application in fractions and standalone expressions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant describes their teacher's method of introducing an "imaginary 1" to eliminate negative exponents, questioning its origin and applicability in various equations.
  • Another participant argues that the "imaginary one" complicates the process unnecessarily, suggesting a simpler approach of moving the base with a negative exponent to the opposite side of the fraction to make the exponent positive.
  • A participant confirms that moving a negative exponent from the denominator to the numerator (or vice versa) effectively changes its sign.
  • There is a discussion about the case of a negative exponent not in a fraction, with one participant proposing that 7^-3 can be expressed as 1/(7^3), which is affirmed by another participant.
  • Another participant introduces a mathematical identity involving exponents, suggesting that the relationship 7^a * 7^b = 7^{a+b} can be extended to negative exponents through manipulation.

Areas of Agreement / Disagreement

Participants express differing views on the necessity and utility of the "imaginary one" in the context of negative exponents. While some agree on the basic principle of moving bases with negative exponents, there is no consensus on the best method to explain or teach this concept.

Contextual Notes

Participants do not fully resolve the implications of using the "imaginary one" in various contexts, nor do they clarify the broader applicability of the methods discussed across different types of equations involving negative exponents.

StupidGenius
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Recently in class (11 math) we explored exponents in negatives. My teacher placed a question on the board and it said

7*1/7^-7
( 7 times 1 over 7 to the power of -7 )

She said that in order to get rid of a negative, we must imagine that there is an imaginary "1" multipliying the fraction.

1/7^-7 *1

Then we must "flip" the fraction around to eliminate the negative exponent. so it will be

7*1*7^7/1

I am a type of person to want to to know why and how stuff are done, especially in the subject of math. I want to know the reasoning behind. Unfortunately i couldn't ask her that day, and then the weekend came and the problem still remains.
So now i know how to get ride of the negative exponent, but will someone explain to me where the imaginary"1" came from and will i be able to do this in any equation where a negative exponent is present?
 
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Personally, I have never heard of the "imaginary one" but from how your teacher explained it, I think it makes the whole problem a lot harder than necessary. So here's my explanation: whenever a number is to a negative exponent, all you have to do is move it to the other side of the fraction to get rid of the negative. If the negative exponent is on the denominator, move it to the numerator and vice versa.
Ex. (2^5)/(3^-4) = (2^5)(3^4)
All I did was move 3^-4 from the denominator to the numerator, thus making it's negative exponent positive.
 
Makes sense, but that "1" still exists in the fraction if (2^5)(3^4)/1. But I see how it is sort of irrelevant. Thank you, now what if the number was not in a fraction and was just simply;
(7^-3)
will this be correct? 1/(7^3)
 
7^-3 = 1/7^3, yes.
 
The basic idea behind this is the equation

7^a 7^b = 7^{a+b}​

We know this is true for a and b both positive integers, so we'd like to make it true for other values too.

Put b=0 and the only solution is 7^0 = 1. Then put b=-a to see that

7^{-a} = \frac{7^0}{7^a} = \frac{1}{7^a}​
 

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