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Math Help - Rational Exponents

  1. #1
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    Rational Exponents

    1) Simplify:

    I know I can split it up like below, but I'm finding it hard to remember where to go next.

    It's obvious that both 9 and 15 are factors of 45, but, in terms of exponents, I can't remember the order of operations from here.

    2) Simplify:

    I know that a^(1/n) = nth root of a, but I can't seem to get the correct answer.

    Thanks in advance.
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  2. #2
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    Re: Rational Exponents

    More important than 9 and 15 being factors of 45 is that 3 and 5 are factors of all three numbers: 9= 3^2, 15= 3(5), and 45= (3^2)(5). So [tex](45)^{1/3}= ((3^2)(5))^{1/3}= (3^{2/3})(5^{1/3}), 9^{3/4}= (3^2)^{3/4}= 3^{3/2}, and 15^{3/2}= (5^{3/2})(3^{3/2}).

    So \frac{45^{1/3}}{(9^{3/4})(15^{3/2})}= \frac{(3^{2/3})(5^{1/3})}{(3^{3/2})(3^{3/2})(5^{3/2})}.

    Can you do it now?
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  3. #3
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    Re: Rational Exponents

    Yes, thanks for the help.

    Could someone point me in the right direction for 2?
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  4. #4
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    Re: Rational Exponents

    Quote Originally Posted by Fratricide View Post
    Could someone point me in the right direction for 2?
    \frac{1}{\sqrt{x-1}}+\sqrt{x-1}=\frac{x}{\sqrt{x-1}}=\frac{x\sqrt{x-1}}{x-1}
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  5. #5
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    Re: Rational Exponents

    If you are asking "How can I look at this problem and instantly know the answer without doing any calculation at all?", I can't help you because I certainly cannot do that. But in about the fourth or fifth grade I learned that to add fractions, you need to get a "common denominator". To add \frac{1}{\sqrt{x- 1}}+ \sqrt{x-1}= \frac{1}{\sqrt{x- 1}}+ \frac{\sqrt{x-1}}{1}, you need the "common denominator" which, obviously, is \sqrt{x- 1}. Multiplying both numerator and denominator of \frac{\sqrt{x- 1}}{1} by \sqrt{x- 1} gives
    \frac{1}{\sqrt{x- 1}}+ \frac{x- 1}{\sqrt{x- 1}}
    Last edited by HallsofIvy; August 11th 2013 at 05:43 AM.
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  6. #6
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    Re: Rational Exponents

    I understand it up to there, but after that I'm not sure what to do.

    That is, I can't grasp how Plato got from here:

    Quote Originally Posted by Plato View Post
    \frac{x}{\sqrt{x-1}}
    To here:

    Quote Originally Posted by Plato View Post
    \frac{x\sqrt{x-1}}{x-1}

    Note: The answer in the textbook is: x(x-1)^-1​.
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  7. #7
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    Re: Rational Exponents

    Quote Originally Posted by Fratricide View Post
    I understand it up to there, but after that I'm not sure what to do.
    That is, I can't grasp how Plato got from here:

    To here:

    Note: The answer in the textbook is: x(x-1)^-1​.
    Note that answer is incorrect.

    \frac{x}{\sqrt{x-1}} can be written as x(x-1)^{-1/2}

    But in general: \frac{x}{\sqrt{x-1}}=\frac{x}{\sqrt{x-1}}\cdot\frac{\sqrt{x-1}}{\sqrt{x-1}}=\frac{x\sqrt{x-1}}{x-1}

    Have a look at this.
    Last edited by Plato; August 11th 2013 at 02:56 PM.
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