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Basic algebra/Power laws equation

  1. Jul 4, 2010 #1
    ok so im working through a problem,got part way through and realised i couldnt remeber what to do with power rules,if someone could quickly remind me of the rules!

    the question is differentiate

    ƒ(x) = 1/x^3

    Using

    if ƒ(x) = x^n then ƒ'(x) = nx^n-1

    and the book answer given is
    ƒ(x) = 1/x^3 = x^-3

    so the derivative is ƒ'(x) = -3x^-4

    now i can understand all but one part of this

    the part which is how to get from

    1/x³ to x^-³

    or why

    1/xⁿ becomes x-ⁿ

    i remember is one of the basic power laws,just cannot remember how or why its so.
    could someone remind me please!
     
    Last edited: Jul 4, 2010
  2. jcsd
  3. Jul 4, 2010 #2
    Part a. x^n * 1/x^n = 1.
    To see this, notice that the left hand member is the same as
    x^n/x^n, and any nonzero number divided by itself is equal to one.
    Part b. x^n * x^-n = 1.
    To see this, use the rule of exponents that says "when multiplying like bases, add the exponents". So you will have x^(n + -n), which is equal to x^(n - n) = x^0 = 1.
    Now look at what changed from part a to part b. I replaced the "1/x^n" in part a with "x^-n" in part b, and the result was unchanged, so the two expressions are equal.

    A more formal approach (using inverse operations, etc) will probably be posted if that's what you're looking for.
     
  4. Jul 4, 2010 #3
    The notations are equivalent:

    [tex]b^{-p}\equiv\frac{1}{b^p}[/tex]

    The left hand side and the right hand side of this equation are two different ways of writing the same thing.
     
  5. Jul 4, 2010 #4
    For the field of real numbers with addition and multiplication as usually defined:

    [tex]b^{p+q}=b^pb^q[/tex]

    [tex]b^{pq}=(b^p)^q[/tex]

    [tex]b^{-1}=\frac{1}{b}=\frac{1}{b^1}[/tex]

    [tex]b^{p+(-q)}=b^{p-q}=\frac{b^p}{b^q}=b^p(b^q)^{-1}=b^pb^{-q}[/tex]

    [tex]b^{p/q}=\sqrt[q]{b^p}[/tex]

    where -x means the inverse of x with respect to the operation of addition (the additive inverse of x), so that x-x means x+(-x) = 0, and 1/x = x-1 means the inverse of x with respect to the operation of multiplication (the multiplicative inverse of x), so that x/x = xx-1 = 1. These rules are general except that 0-1 (and hence any negative power of 0) is not defined.

    If p and q are whole numbers, bp can be interpreted as multiply p b's together, and bp-q multiply p b's together with q instances of the multiplicative inverse of b, with the convention that b0 = 1. The rest of the real numbered powers "fill in the gaps" between rational powers (fractional powers). I don't yet know how that filling in of the gaps is formally defined, but I'm sure there are lots of people here who do!
     
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