Results 1 to 7 of 7

Math Help - is x to the power of 1/3 the same as x to the power of 2/6? see inside please

  1. #1
    Junior Member
    Joined
    Apr 2008
    Posts
    65

    is x to the power of 1/3 the same as x to the power of 2/6? see inside please

    x^(1/3) ... is it the same as x^(2/6) ?

    lets consider x < 0

    x^(1/3) is okay

    (-8)^(1/3) = -2

    then

    x^(1/3) = (x^2)^(1/6) is okay since (-8)^2 = 64^(1/6) ((but this = 2))

    and finally

    x^(1/3) = (x^(1/6)^2) is not okay since (-8)^(1/6) = undefined


    does anyone have an explanation, idea or comment about this? my math teacher brought this up to me today
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by finch41 View Post
    x^(1/3) ... is it the same as x^(2/6) ?

    lets consider x < 0

    x^(1/3) is okay

    (-8)^(1/3) = -2

    then

    x^(1/3) = (x^2)^(1/6) is okay since (-8)^2 = 64^(1/6) ((but this = 2))

    and finally

    x^(1/3) = (x^(1/6)^2) is not okay since (-8)^(1/6) = undefined


    does anyone have an explanation, idea or comment about this? my math teacher brought this up to me today
    It depends if you are strict...I can see it from both ways...since you haev a brief indtroduction of i..but then it is elminated...so I would say they are the same..
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    652
    Thanks
    2
    Awards
    1
    Well, lets see, we have:

    x^{\frac{1}{3}}

    And we want to know if:

    x^{\frac{2}{6}}

    is the same.

    Well, let's use the root versions of these:

    \sqrt[3]{x}

    and

    \sqrt[6]{x^2}

    Ok, let's use 8:

    \sqrt[3]{8} = 2

    \sqrt[6]{64} = ?

    Well, we are going to assume that it is 2, well, what's 2 to the 6th power?

    2^6 = 2*2*2*2*2*2 = 4*2*2*2*2 = 8*2*2*2 = 16*2*2 = 32*2 = 64

    Looks like they're the same to me, now let's try -8

    \sqrt[3]{-8} = -2

    \sqrt[6]{(-8)^2} = 2

    Looks like they're only the same for all |x|.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Apr 2008
    Posts
    65
    good posts guys

    my teacher's point was that he was not sure if they are the same function since it can be seen that the domains are uncertain to be the same

    if the domain of two functions is not the same then they are not the same function right
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    652
    Thanks
    2
    Awards
    1
    If you restrict the domain to \mathbb{N} then they are the exact same function. When you cross into the negatives, then things get a bit funky.

    \{x^n = x^{\frac{2}{2}n}|\forall x\in\mathbb{N}\}
    Last edited by Aryth; May 1st 2008 at 01:22 PM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Apr 2008
    Posts
    65
    could you explain what you mean by funky and by what you wrote there
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    652
    Thanks
    2
    Awards
    1
    What I mean by funky is that the two functions have opposite signs once your domain goes into the negative integers.

    And the bottom just says that x to some power n is equal to x to the power of (2/2)n for all x in the natural number line (0,1,2,3,...).

    As we saw, when n=\frac{1}{3} and x=8 and \frac{2}{2}n = \frac{2}{6}, then:

    x^n = x^{\frac{2}{2}n} for all x > 0.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Power set of B
    Posted in the Discrete Math Forum
    Replies: 10
    Last Post: November 4th 2010, 01:00 PM
  2. Power
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: October 16th 2009, 03:06 PM
  3. Exponents power to a power
    Posted in the Algebra Forum
    Replies: 3
    Last Post: February 27th 2009, 12:39 AM
  4. Power mean
    Posted in the Calculus Forum
    Replies: 2
    Last Post: August 30th 2008, 05:16 AM
  5. Replies: 10
    Last Post: April 18th 2008, 10:35 PM

Search Tags


/mathhelpforum @mathhelpforum