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Thread: Imagining something to the power of fraction(1/2,1/3,3/4) on the number line

  1. #1
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    Imagining something to the power of fraction(1/2,1/3,3/4) on the number line

    Hi,

    It took me five minutes to learn latex but finally my question is here.

    $\displaystyle x^2 = y$
    $\displaystyle x = \sqrt{y}$
    $\displaystyle y^\frac{1}{2}=x$

    I am able to imagine$\displaystyle x^2$ on the number line in my mind How to imagine something to a power of 1/2 or any fraction on the number line.

    Thank you
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  2. #2
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    Re: Imagining something to the power of fraction(1/2,1/3,3/4) on the number line

    You say "finally my question is here", but I see know question!

    You say "I am able to imagine on the number line in my mind". What do you mean by that? If x= 2 then $\displaystyle x^2= 4$, a point on the number line, but for general x, $\displaystyle x^2$ can be any point representing a non-negative number.

    As for $\displaystyle sqrt{x}= x^{1/2}$, if x= 4 then $\displaystyle \sqrt{x}= 2$ but, for any non-negative x, $\displaystyle \sqrt{x}$ can, again, be any point representing a non-negative number.
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    Re: Imagining something to the power of fraction(1/2,1/3,3/4) on the number line

    Hi,


    $\displaystyle x^3 = x \times x \times x$
    $\displaystyle x^\frac{1}{2}$ = ?? represented in the terms of product of x

    I know this is basics but please provide the imaginative way to represent this
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  4. #4
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    Re: Imagining something to the power of fraction(1/2,1/3,3/4) on the number line

    $\displaystyle x^3 = x \times x \times x = y $ where y >= x >= 1
    $\displaystyle x^\frac{1}{n} = ??? = t $ then 1 <= t <= x and t to the power of the fraction denominator = x
    example $\displaystyle \sqrt{2} = 2^\frac{1}{2} = 1.414^2 \approx 2 $

    I am still not able to get how the reverse of an operation can be represented in the same form

    multiplication is the reverse of division. Here I see the reverse of exponentiation can be represented using exponentiation itself but the operation is different and its not exponentiation at all

    Exponentiation - Wikipedia, the free encyclopedia.

    nth root - Wikipedia, the free encyclopedia

    In calculus, roots are treated as special cases of exponentiation, where the exponent is a fraction:

    $\displaystyle \sqrt[n]{x} \,=\, x^{1/n}$ = why is this ??
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