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Math Help - decimal exponents and roots, whats the big idea?

  1. #1
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    decimal exponents and roots, whats the big idea?

    what is the concept around an exponent or root that is a decimal.

    its cool for example 125^2 is telling me "times me by myself"
    125^3 is telling me "times me by myself and do it again!"


    but what is 125^2.5 telling me to do?
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by flacck42 View Post
    what is the concept around an exponent or root that is a decimal.

    its cool for example 125^2 is telling me "times me by myself"
    125^3 is telling me "times me by myself and do it again!"


    but what is 125^2.5 telling me to do?
    Some of this is easy, some of it isn't.

    Decimals that can be readily turned into fractions have an immediate interpretation. For example, you can show that
    a^{1/n} = \sqrt[n]{a}

    So your example
    125^{2.5} = 125^{5/2} = \sqrt{125^5}

    If the decimal is something like \sqrt{2} (ie. irrational) I am much more hazy about the interpretation. We can, however, do this:
    a^{\sqrt{2}} = a^{1.41421...}

    = a^{1 + 0.4 + 0.01 + 0.004 + 0.0002 + 0.00001 + ...}

    = (a^1)(a^{4/10})(a^{1/100})(a^{4/1000})(a^{2/10000})(a^{1/100000})...

    But that's still pretty messy to work with.

    -Dan
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by flacck42 View Post
    what is the concept around an exponent or root that is a decimal.

    its cool for example 125^2 is telling me "times me by myself"
    125^3 is telling me "times me by myself and do it again!"


    but what is 125^2.5 telling me to do?
    fractional exponents get a little weird.

    Note that 125^{2.5} = 125^{5/2}

    now what this says is "times me by myself and do it again and again and again, and while you at it, take my square root (but that's if you interpret me as \left( 125^5\right)^{1/2}). If you interpret me as  \left( 125^{1/2} \right)^5, then take my square root and then raise me to the 5th power"

    thus, 125^{2.5} = 125^{5/2} = \sqrt{125^5} = \left( \sqrt{125} \right)^5

    if the power is irrational (cannot be expressed as a fraction, then, you need your calculator)

    see post #3 here for more info


    EDIT: Thanks for letting me waste my time answering this, Dan
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Jhevon View Post
    EDIT: Thanks for letting me waste my time answering this, Dan
    Always glad to be of service!

    (Hey Jhevon, if you want to see a reaction, give some +rep to TPH. I did earlier and he said it gave him like 50 points. I'm curious to see what kind of muscle you have in that department.)

    -Dan
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by topsquark View Post
    Always glad to be of service!

    (Hey Jhevon, if you want to see a reaction, give some +rep to TPH. I did earlier and he said it gave him like 50 points. I'm curious to see what kind of muscle you have in that department.)

    -Dan
    hehe, you know you caused him to pass you

    i will try, this thing stops me from adding rep sometimes. i've been trying to add rep to Krizalid forever (i just tried to add rep to you and it wouldn't allow me to)

    TPH went offline, hopefully next time he logs on he remembers how many points he had before i added rep
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  6. #6
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Jhevon View Post
    hehe, you know you caused him to pass you

    i will try, this thing stops me from adding rep sometimes. i've been trying to add rep to Krizalid forever
    Good thing that I'm not concerned with it then. You need to "spread it around" some before you can go back and give rep to the same person again. (Though I wonder if you can + rep then - rep the same person?)

    -Dan
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  7. #7
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    question two

    should i rack my brain mulling over your concepts or would they come to me eventually (through osmosis?) and chillax a bit?

    would i be in better position to sink these concepts after i learned calculus?
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  8. #8
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by flacck42 View Post
    question two

    should i rack my brain mulling over your concepts or would they come to me eventually (through osmosis?) and chillax a bit?
    definitely rack your brain. the osmosis takes care of itself. you'd find that sometimes after racking your brain for a long time, you will give up and go to something else, when suddenly, BAM! with no warning, it comes to you. (your subconscious will get the message that figuring this out is important to you because you were racking your brain, and so it works on it while you're not noticing)

    would i be in better position to sink these concepts after i learned calculus?
    you should know concepts like these before going to calculus, you do not want to be in calculus if you do not have basic proficiency in algebra.
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  9. #9
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    got it:

    125^2.5 says

    split 125 into two equal parts that when multiplied equals 125:
    11.2*11.2=125

    take one of those parts (11.2) and multiply that by 125 and 125 again.


    likewise 125^2.75 says

    split 125 into four parts that when multiplied together equals 125
    3.3*3.3*3.3*3.3=125

    take three of those parts 3.3*3.3*3.3=37.4 and multiply by 125 and 125 again.



    im going to watch cartoons.
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  10. #10
    Super Member angel.white's Avatar
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    Quote Originally Posted by Jhevon View Post
    hehe, you know you caused him to pass you

    i will try, this thing stops me from adding rep sometimes. i've been trying to add rep to Krizalid forever (i just tried to add rep to you and it wouldn't allow me to)

    TPH went offline, hopefully next time he logs on he remembers how many points he had before i added rep
    I made a calculator to determine your reputation (I assume the same weights are default for all vbulletin sites, and that neither this site's weight nor the site I wrote it for's weight was modified. If this is not the case, then the calculator will tell you the wrong answer)

    calculate your power of reputation - Starcraft Dream

    The exact specifications for determining can be found here, but you need admin CP access to find the numbers vBulletin Manual

    Or, if you plug your numbers into the equation, and they come out to roughly 50 points, then it is probably accurate.
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  11. #11
    Junior Member qspeechc's Avatar
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    What about 5^(sqrt(2))?
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  12. #12
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by qspeechc View Post
    What about 5^(sqrt(2))?
    Do you mean how to calculate it?

    Probably the simplest thing to do is to get the decimal representation of \sqrt{2} = 1.41421....

    Thus
    5^{\sqrt{2}} = 5^{1.41421...} = (5^1)(5^{0.4})(5^{0.01})(5^{0.004})(5^{0.0002})(5^  {0.00001})...

    = 5 \cdot 5^{4/10} \cdot 5^{1/100} \cdot 5^{4/1000} \cdot 5^{2/10000} \cdot 5^{1/100000} \cdot ~...

    So you can calculate the first several terms in the product and get at least a decent approximation to it. (Note as the series progresses to smaller and smaller fractions 5^n approaches 1.) But you can't get an exact value for it.

    Plugging it into my calculator gives 5^{\sqrt{2}} \approx 9.73852

    -Dan
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