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Math Help - simplifying radicals

  1. #1
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    please help me understand these evil radicals!

    okay, so i'm learning how to simplify radicals and i understand the rules for the most part of simplifying but here is an example of what is confusing me:

    Question:
    X^2 Z^2 ^6(1/X^17)

    answer:
    Z^2/X ^6(x)

    okay because i don't know how to do the fancy stuff on the computer, this isn't how the actual problem looks. The carrots ^ stand for the exponents so the number after them is the exponent and the ^6 is the index of the radical and the numbers in the parenthesis (...) are the numbers in the radical so insead of parethesis there really should be one of those square root thingys.

    So i obviously don't need you to solve this problem for me becuase i have the answer BUT i don't know how to get that answer at all and im just so confused how they did it so i just need someone to explain to me how this answer came about.

    Side Notes:

    The three rules to simplifying are:
    1.) All exponents inside ^n (...) are positive and less then n
    2.) No Common factor can be removed from n and all exponents.
    3.) The denominator is rationalized.

    okay so just remember that the parenthesis aren't really parenthesis.. It is a radical square root symbol instead. and the 1/X^17 means that 1 is on top and the X^17 is on the denominator but all of it is to the 6th square root.. or however you would say it.
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  2. #2
    Site Founder Math Help's Avatar
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    simplify radicals

    Quote Originally Posted by AlizzleShizzle
    Question:
    X^2 Z^2 ^6(1/X^17)

    answer:
    Z^2/X ^6(x)
    So the problem is:

    x^2*z^2*\sqrt[6]{\frac{1}{x^{17}}}

    we must get the x out of the denominator under the radical. therefore we need at least an x^{18} in the numerator. We can multiply by "1" creatively.
    In this case we divide by x^3 and multiply by \sqrt[6]{x^{18}}.

    so we have:

    \frac{x^2*z^2}{x^3}*\sqrt[6]{\frac{x^{18}}{x^{17}}}

    I think you can take it from there.
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  3. #3
    Site Founder Math Help's Avatar
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    Lightbulb Fancy math

    By the way if you want to learn Latex check this out http://www.mathhelpforum.com/math-he...read.php?t=266
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  4. #4
    MHF Contributor
    Joined
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    Here is one way.

    I will solve it even if you said I don't have to, because you know the answer.
    But then, you also said you don't know how to get the answer (even if you know thw answer), so I will show you one solution.

    ---------
    Question:
    X^2 Z^2 ^6(1/X^17).
    Per your explanations, that will be
    x^2 z^2 sixthroot[1/ (x^17)]

    answer:
    Z^2/X ^6(x)
    That will be
    [(z^2)/x]*sixthroot(x)
    Or, [z^2 sixthroot(x)] / x

    ---------------------------------
    x^2 z^2 sixthroot[1/ (x^17)]

    See that denominator x^17 in the radicand? We can bring a portion of that outside the radical sign. This portion must be "x raised to a multiple of 6", because 6 is the index of the radical. We see that 12 is the maximum we can get from 17, so,
    = x^2 z^2 sixthroot[1/ (x^12 x^5)]
    = x^2 z^2 {[1/(x^2)] sixthroot[1/(x^5)]}
    = [(x^2 z^2) / (x^2)] sixthroot[1/(x^5)]
    = [z^2] sixthroot[1/(x^5)]

    Now see that denominator x^5 in the radicand? We can get that outside the radical sign if we can make it x^6.
    No problem, we multiply it by x, so (x^5)(x) = x^6. We do that to the denominator, we have to do it to the numerator also so that we won't change the original value of the radicand, which is 1/(x^5). So the new radicand now will be (1/ x^5)(x/x) = x/(x^6). Hence,

    = [z^2] sixthroot[x/(x^6)]
    = [z^2][(1/ x) sixthroot(x)]
    = [(z^2)/x]sixthroot(x) -----------answer.

    ---------------
    A shorter version of the above is if we make the x^17 into x^18 right away, because 18 is 3*6.

    x^2 z^2 sixthroot[1/ (x^17)]

    = x^2 z^2 sixthroot[(1/ (x^17))(x/x)]
    = x^2 z^2 sixthroot[x/ (x^18)]
    = x^2 z^2 {[1/(x^3)]sixthroot[x]}
    = [(x^2 z^2) / (x^3)]sixthroot(x)
    = [(z^2)/x]sixthroot(x) ------------answer.
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