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Why does b^(m/n) = sqrt(n,m)?

  1. Oct 2, 2003 #1
    Why does b^(m/n) = (nsqrt(b))^m?

    Hi, as the subject says, why does b^(m/n) = (n√b)^m?

    I don't understand how you can multiply a number by itself less than one times.

    Thanks.

    EDIT: Finally GOT IT RIGHT.
     
    Last edited: Oct 3, 2003
  2. jcsd
  3. Oct 2, 2003 #2
    b^(m/n)= nã(b^m) "the nth root of b to the m power"
    you could also write (nãb)^m
    Aaron
     
  4. Oct 2, 2003 #3
    I meant that but I was just thinking about too many things. It's been fixed. I'm asking for an explanation of why that is true.
     
  5. Oct 2, 2003 #4
    Re: Why does b^(m/n) = (nsqrt(m))^n?

    I don't know that it does. 31/2=(2[squ]1)1=2?
     
  6. Oct 3, 2003 #5

    HallsofIvy

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    The stupid errors just keep piling up don't they!

    I take it you mean: "Why is bm/n= n &radic (b)m?"

    Let's start with "I don't understand how you can multiply a number by itself less than one times."

    You can't. bn is defined as "multiply b by itself n times" only if n is a positive integer (counting number).

    However, in that simple situation, we quickly derive the very useful "laws of exponents": bmbn= bm+n and (bm)n= bnm.

    We then define bx for other number so that those laws remain true.

    For example, IF the laws of exponents are to be true for x= 0, then we must have bn= bn+0= bnb0. As long as b is not 0 we can divide both sides of the equation by bn to bet b0= 1. That is, we MUST define b0= 1 or the laws of exponents will no longer hold.

    Now we can see that bn+(-n)= b0= 1. If the laws of exponents are to hold for negative exponents as well, we must have bnb-n= bn+(-n)= 1 or, again dividing both sides of the equation by bn, b-n= 1/bn.

    Finally, if (bm)n= bmn is to be true for all numbers, we must have (b1/m)m= b1= b. Since n &radic (b) is define as "the positive number whose nth power is b, we must define b1/m= m &radic (b).
     
    Last edited: Oct 3, 2003
  7. Oct 3, 2003 #6
    Thanks HallsOfIvy, your explanation was very clear.

    And yes, the errors kept piling up! I have fixed everything but the subject (I don't believe it can be changed. Am I wrong?) so if anyone wants to read it in the future it should make sense.
     
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