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Math Help - Radicals and exponents

  1. #1
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    Question Radicals and exponents

    Hi, I am stuck on this problem, please help.

    ((x^2+1)^(1/2) - x^2(x^2-1)^(-1/2)) / (x^2-1)

    The book says 1/(x^2+1)^3/2

    While trying different things I figured out the rule that the square root of a divided by a = 1/ square root of a. There must a rule I could use here. My answer always has x^2 in the numerator.

    Thanks!
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  2. #2
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    Quote Originally Posted by dwatkins741 View Post
    Hi, I am stuck on this problem, please help.

    ((x^2+1)^(1/2) - x^2(x^2-1)^(-1/2)) / (x^2-1)

    The book says 1/(x^2+1)^3/2

    While trying different things I figured out the rule that the square root of a divided by a = 1/ square root of a. There must a rule I could use here. My answer always has x^2 in the numerator.

    Thanks!
    Dear dwatkins741,

    So the book says that,

    \frac{(x^2+1)^{\frac{1}{2}}-x^2(x^2-1)^{-\frac{1}{2}}}{x^2-1}=\frac{1}{(x^2+1)^{3/2}}

    But notice that if x=0 then, \frac{(x^2+1)^{\frac{1}{2}}-x^2(x^2-1)^{-\frac{1}{2}}}{x^2-1}=-1

    But \frac{1}{(x^2+1)^{3/2}}=1

    Hence there is a mistake in this problem,

    \frac{(x^2+1)^{\frac{1}{2}}-x^2(x^2-1)^{-\frac{1}{2}}}{x^2-1}\neq\frac{1}{(x^2+1)^{3/2}}

    Hope this will help you.
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  3. #3
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    Red face Correction

    I sincerely apologize. I had two signs wrong when I typed the problem. Please take another look at it.

    ((x^2+1)^(1/2) - x^2(x^2+1)^(-1/2)) / (x^2+1)

    Sorry, thanks.
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  4. #4
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    Quote Originally Posted by dwatkins741 View Post
    I sincerely apologize. I had two signs wrong when I typed the problem. Please take another look at it.

    ((x^2+1)^(1/2) - x^2(x^2+1)^(-1/2)) / (x^2+1)

    Sorry, thanks.
    Dear dwatkins741,

    Don't mention it. Now you can solve the problem. First multiply the denominatior and the numerator by, (x^2+1)^{\frac{1}{2}}. Hope you can continue.
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  5. #5
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    Quote Originally Posted by dwatkins741 View Post
    ((x^2+1)^(1/2) - x^2(x^2+1)^(-1/2)) / (x^2+1)
    Go this way: let a = x^2 + 1 ; then:
    [sqrt(a) - x^2/sqrt(a)] / a : remember that k^(-p) = 1/k^p
    = [sqrt(a)sqrt(a) - x^2] / [a sqrt(a)]
    = (a - x^2) / [a^1 a^(1/2)]
    = (a - x^2) / a^(3/2) : remember that k^p k^q = k^(p+q)

    Finish it by substituting back in...
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