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Math Help - Fractional Expontent of a Binomial - Exponent Rules Question

  1. #1
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    Fractional Expontent of a Binomial - Exponent Rules Question

    Hello,


    To evaluate the expression (x-2)^1/2 the next steps listed are x^(-2)(1/2)
    then x^-1 and finally 1/x. I am fine with the final two steps.

    Please help me see how or what rules are used to go from the first to second step. Is this just a basic rules I am forgetting?


    So in summary how does (x-2)^1/2 become x^(-2)(1/2)

    Thank you.
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  2. #2
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    Re: Fractional Expontent of a Binomial - Exponent Rules Question

    Quote Originally Posted by joseclar View Post
    Hello,


    To evaluate the expression (x-2)^1/2 the next steps listed are x^(-2)(1/2)
    then x^-1 and finally 1/x. I am fine with the final two steps.

    Please help me see how or what rules are used to go from the first to second step. Is this just a basic rules I am forgetting?


    So in summary how does (x-2)^1/2 become x^(-2)(1/2)

    Thank you.
    1. An example:

    \left(b^3\right)^4 = \left(b^3\right) \cdot \left(b^3\right) \cdot \left(b^3\right) \cdot \left(b^3\right) = \left(b\right)^{3+3+3+3} = b^{3 \cdot 4}

    So if you take a power to a power you have to multiply the exponents.

    2. In general:

    \left(b^a\right)^c = b^{a\cdot c} = \left(b^c\right)^a

    3. Apply this rule to your question.
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    Re: Fractional Expontent of a Binomial - Exponent Rules Question

    tHANK YOU for the reply. I am aware of the rule for multiplying exponents. My question is how does (x-2)^1/2 become x^-2(1/2). Thanks.
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    Re: Fractional Expontent of a Binomial - Exponent Rules Question

    Quote Originally Posted by joseclar View Post
    My question is how does (x-2)^1/2 become x^-2(1/2). Thanks.
    Why do you think it does? In what context does it appear?
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  5. #5
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    Re: Fractional Expontent of a Binomial - Exponent Rules Question

    Quote Originally Posted by joseclar View Post
    tHANK YOU for the reply. I am aware of the rule for multiplying exponents. My question is how does (x-2)^1/2 become x^-2(1/2). Thanks.
    ... by typo!

    Quote Originally Posted by joseclar View Post
    Hello,


    To evaluate the expression (x-2)^1/2 the next steps listed are x^(-2)(1/2)
    then x^-1 and finally 1/x. I am fine with the final two steps. <--- who has listed these steps?

    Please help me see how or what rules are used to go from the first to second step. Is this just a basic rules I am forgetting?


    So in summary how does (x-2)^1/2 become x^(-2)(1/2)

    Thank you.
    1. I assumed that there was a typo in the original term and that actually was meant \left(x^{-2} \right)^{\frac12}

    2. If you re-arrange the given term

    (x-2)^{\frac12} = \sqrt{x-2}

    you can see that

     \sqrt{x-2} = \frac1x~\implies~x^3-2x^2-1=0

    has only one real solution at x = \frac{\sqrt[3]{172-12 \cdot \sqrt{177}}}{6}+\frac{\sqrt[3]{172+12 \cdot \sqrt{177}}}{6} +\frac23

    To answer your question So in summary how does (x-2)^1/2 become x^(-2)(1/2): Only once - and in all other cases this transformation is wrong.
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