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> What is the reason that when different powers of the same quantity are multiplied together their exponents are added? As a math professor, I think this is a

> What is the reason that when different powers of the same quantity are multiplied together their exponents are added? As a math professor, I think this is a great question. Students learn that math is about manipulating formulas and equations, or about excessive formalities. But being able to explain simple arithmetic facts in clear and plain English is often neglected, and is of the utmost value.

hammock 15y ago
Who here can come up with the most concise explanation?
- cousin_it 15y ago
  Me, me! Let me try! X^N * X^M = N copies of X, multiplied by M copies of X = N+M copies of X multiplied together = X^(N+M)
  - baddox 15y ago
    That's not very rigorous, especially with fractional (or God forbid, irrational) exponents.
    - cousin_it 15y ago
      Yeah, and I'd be even more interested in a similarly intuitive explanation for the case of complex exponents ;-)
      - baddox 15y ago
        That would be difficult, since a formal construction of even the real numbers is a somewhat advanced (3rd or 4th year college mathematics) topic. I forget the details, but I believe a^n for real a and complex n is formally defined using the exponential function (e^x).
        jules 15y ago
        At least in the Netherlands, construction of the reals (for example from rational Cauchy sequences) is standard 1st semester stuff. Understanding reals is required or provides a good source of examples for virtually all mathematics courses, so I can't imagine how some universities teach mathematics without it.
    - jerf 15y ago
      Given that the exam is from 1869 for entrance into college, I suspect that's basically what they are looking for.
      - Queue29 15y ago
        Yes, because irrational numbers wouldn't be discovered until the 1900's....
        bbrizzi 15y ago
        Actually one of the ways to prove that requires the fact that R completes Q. The concept of completeness was introduced by Cauchy in the 19th century and it might not have been completely popularised in 1869.
    - impendia 15y ago
      Indeed this is informal and could be made more rigorous, but even at the highest level of rigor, I think it's most natural to do integral exponents and then rational. Indeed you have to construct the integers before you can construct the rationals.
- deleted 15y ago
  [deleted]
- baby 15y ago
  (a^n) * (a^m) = (a*a*a*a*a) * (a*a*a*a*a*a*a) ^ ^ n times m times I don't see what is simpler than this, It comes from the basic definition of what a power is.
  - dhconnelly 15y ago
    this, with the associative law
    - baby 15y ago
      I could have wrote it without ( ) and I don't think we learn what associative means in highschool.
- twelvechairs 15y ago
  The process of explanation by example (though I agree with others that it is really intrinsic in the meaning of a power - id like to hear impendia's explanation): a^(n+1)=a^n * a a^(n+2)=a^n * a^2 a^(n+3)=a^n * a^3 a^(n+0)=a^n * a^0 therefore a^(n+m)=a^n * a^m
  - impendia 15y ago
    > id like to hear impendia's explanation I personally like cousin_id's the best, as it is (IMHO) the simplest.
- lolcraft 15y ago
  Another "proof": x^n = exp((log x)n). By definition, exp(n+m)=exp(n)exp(m). By definition, a(b+c) = ab + ac (for the log x thing). QED. By the way, (the infamous calculus book) Baby Rudin has the poor reader show this property holds in exponentiation for reals, starting with integers and via rationals, as an exercise on its first chapter. Insane difficulty for me, even though the author practically holds your hand along the way! Cool read, though.
fuzionmonkey 15y ago
Do you actually find college students that don't know what an exponent is? I feel like anybody with a basic understanding of what an exponent represents should be able to explain why you add exponents.
- impendia 15y ago
  They have a working knowledge of it, with very rare exceptions -- but I find that students are taught math as a bunch of rules and not all of them are comfortable giving explanations.
  - the-cakeboss 15y ago
    Yeah, that is what I perceive as the greateset failing of modern math education. More emphasis on memorization rather than a deep understanding.
    - slowpoke 15y ago
      The thing is, most fields that apply math do not need to "deeply understand" math. I think it is highly arrogant to imply that everyone needs a deep understanding of math to use the tools it provides. For most people which aren't mathematicians, it's perfectly fine to just be able to apply formula. They don't gain anything from a deep understanding of the matter. It's the same as libraries in programming, really. You don't need to understand how a library works, you need to know what it does and how to use it. And in many cases, that's perfectly fine. After all, who would have the time to study the source code[1] of all libraries they use? You'd study that if you are interested, noticed a bug or need to know some specific detail of the implementation, not just because you need to use it. [1] Not to mention those unfree binary libraries which you couldn't inspect even if you wanted.
      - the-cakeboss 15y ago
        A deep understanding allows one to deal with problems that they have not only been explicitly taught, but those they have also never encountered. I don't think your example regarding libraries really applies well here. I think a more appropriate example would be knowing how to do a few things with a library without a real ( or any ) understanding of the language its based upon. I'm no mathematician, nor am I a (professional) programmer for that matter, yet I've seen real benefits going back and really learning some of those topics that were kinda passed over in high school ( so what exactly is sine doing to my numbers? Taylor series? Fourier series? Optimization functions?). Besides, math teaches you to think, memorization doesn't.
baddox 15y ago
That question stood out to me as a particularly bad question. What is the answer supposed to be? I completely understand how multiplication of exponents works, but I have no idea how to describe the "reason." You can give a simple algebraic proof quite easily (especially if we're just dealing with integer exponents), but unless "reason" had a more specific mathematical meaning in that time, it seems like a very vague question to me.
- michaeldhopkins 15y ago
  That's the point -- you don't understand it well enough to explain it.
  - barrkel 15y ago
    I don't think that's the case. As baddox says, it's trivial to show it is true, especially using simplified definitions for exponentiation (i.e. sticking with integer or perhaps rational exponents), but demonstrating truth doesn't tell you about the "reason". Is the question about a philosophical position as to how mathematics relates to God? A "reason" seems to imply a purpose.
    - dlss 15y ago
      Being able to explain proofs intuitively is a valuable way to check how deeply you know them. In this case, the reason that when different powers of the same quantity are multiplied together their exponents are added is because powers are short hand for a series of multiplications: 2^4 == 2 * 2 * 2 * 2 When you multiply 2^4 * 2^4, that is short hand for: 2^4 * 2^4 == 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 == 2^8 Of course you can also prove this using algebra, but the intuitive explanation is IMO more useful for building understanding.
      - baddox 15y ago
        That works for positive integer bases and exponents, but try giving an "intuitive proof" with irrational exponents. Most things in math, even seemingly obvious things in arithmetic, require a lot of shared background knowledge (at least propositional logic, basic set theory, and a construction of the natural numbers) for two people to even converse formally.

hammock 15y ago

Who here can come up with the most concise explanation?

cousin_it 15y ago
Me, me! Let me try! X^N * X^M = N copies of X, multiplied by M copies of X = N+M copies of X multiplied together = X^(N+M)
- baddox 15y ago
  That's not very rigorous, especially with fractional (or God forbid, irrational) exponents.
  - cousin_it 15y ago
    Yeah, and I'd be even more interested in a similarly intuitive explanation for the case of complex exponents ;-)
    - baddox 15y ago
      That would be difficult, since a formal construction of even the real numbers is a somewhat advanced (3rd or 4th year college mathematics) topic. I forget the details, but I believe a^n for real a and complex n is formally defined using the exponential function (e^x).
      - jules 15y ago
        At least in the Netherlands, construction of the reals (for example from rational Cauchy sequences) is standard 1st semester stuff. Understanding reals is required or provides a good source of examples for virtually all mathematics courses, so I can't imagine how some universities teach mathematics without it.
  - jerf 15y ago
    Given that the exam is from 1869 for entrance into college, I suspect that's basically what they are looking for.
    - Queue29 15y ago
      Yes, because irrational numbers wouldn't be discovered until the 1900's....
      - bbrizzi 15y ago
        Actually one of the ways to prove that requires the fact that R completes Q. The concept of completeness was introduced by Cauchy in the 19th century and it might not have been completely popularised in 1869.
  - impendia 15y ago
    Indeed this is informal and could be made more rigorous, but even at the highest level of rigor, I think it's most natural to do integral exponents and then rational. Indeed you have to construct the integers before you can construct the rationals.
deleted 15y ago
[deleted]
baby 15y ago
(a^n) * (a^m) = (a*a*a*a*a) * (a*a*a*a*a*a*a) ^ ^ n times m times I don't see what is simpler than this, It comes from the basic definition of what a power is.
- dhconnelly 15y ago
  this, with the associative law
  - baby 15y ago
    I could have wrote it without ( ) and I don't think we learn what associative means in highschool.
twelvechairs 15y ago
The process of explanation by example (though I agree with others that it is really intrinsic in the meaning of a power - id like to hear impendia's explanation): a^(n+1)=a^n * a a^(n+2)=a^n * a^2 a^(n+3)=a^n * a^3 a^(n+0)=a^n * a^0 therefore a^(n+m)=a^n * a^m
- impendia 15y ago
  > id like to hear impendia's explanation I personally like cousin_id's the best, as it is (IMHO) the simplest.
lolcraft 15y ago
Another "proof": x^n = exp((log x)n). By definition, exp(n+m)=exp(n)exp(m). By definition, a(b+c) = ab + ac (for the log x thing). QED. By the way, (the infamous calculus book) Baby Rudin has the poor reader show this property holds in exponentiation for reals, starting with integers and via rationals, as an exercise on its first chapter. Insane difficulty for me, even though the author practically holds your hand along the way! Cool read, though.

fuzionmonkey 15y ago

Do you actually find college students that don't know what an exponent is? I feel like anybody with a basic understanding of what an exponent represents should be able to explain why you add exponents.

impendia 15y ago
They have a working knowledge of it, with very rare exceptions -- but I find that students are taught math as a bunch of rules and not all of them are comfortable giving explanations.
- the-cakeboss 15y ago
  Yeah, that is what I perceive as the greateset failing of modern math education. More emphasis on memorization rather than a deep understanding.
  - slowpoke 15y ago
    The thing is, most fields that apply math do not need to "deeply understand" math. I think it is highly arrogant to imply that everyone needs a deep understanding of math to use the tools it provides. For most people which aren't mathematicians, it's perfectly fine to just be able to apply formula. They don't gain anything from a deep understanding of the matter. It's the same as libraries in programming, really. You don't need to understand how a library works, you need to know what it does and how to use it. And in many cases, that's perfectly fine. After all, who would have the time to study the source code[1] of all libraries they use? You'd study that if you are interested, noticed a bug or need to know some specific detail of the implementation, not just because you need to use it. [1] Not to mention those unfree binary libraries which you couldn't inspect even if you wanted.
    - the-cakeboss 15y ago
      A deep understanding allows one to deal with problems that they have not only been explicitly taught, but those they have also never encountered. I don't think your example regarding libraries really applies well here. I think a more appropriate example would be knowing how to do a few things with a library without a real ( or any ) understanding of the language its based upon. I'm no mathematician, nor am I a (professional) programmer for that matter, yet I've seen real benefits going back and really learning some of those topics that were kinda passed over in high school ( so what exactly is sine doing to my numbers? Taylor series? Fourier series? Optimization functions?). Besides, math teaches you to think, memorization doesn't.

baddox 15y ago

That question stood out to me as a particularly bad question. What is the answer supposed to be? I completely understand how multiplication of exponents works, but I have no idea how to describe the "reason." You can give a simple algebraic proof quite easily (especially if we're just dealing with integer exponents), but unless "reason" had a more specific mathematical meaning in that time, it seems like a very vague question to me.

michaeldhopkins 15y ago
That's the point -- you don't understand it well enough to explain it.
- barrkel 15y ago
  I don't think that's the case. As baddox says, it's trivial to show it is true, especially using simplified definitions for exponentiation (i.e. sticking with integer or perhaps rational exponents), but demonstrating truth doesn't tell you about the "reason". Is the question about a philosophical position as to how mathematics relates to God? A "reason" seems to imply a purpose.
  - dlss 15y ago
    Being able to explain proofs intuitively is a valuable way to check how deeply you know them. In this case, the reason that when different powers of the same quantity are multiplied together their exponents are added is because powers are short hand for a series of multiplications: 2^4 == 2 * 2 * 2 * 2 When you multiply 2^4 * 2^4, that is short hand for: 2^4 * 2^4 == 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 == 2^8 Of course you can also prove this using algebra, but the intuitive explanation is IMO more useful for building understanding.
    - baddox 15y ago
      That works for positive integer bases and exponents, but try giving an "intuitive proof" with irrational exponents. Most things in math, even seemingly obvious things in arithmetic, require a lot of shared background knowledge (at least propositional logic, basic set theory, and a construction of the natural numbers) for two people to even converse formally.