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Math Help - Real Analysis Problem

  1. #1
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    Real Analysis Problem

    For a real number b>0 and a real number x, define b^x, and show b^xb^y = b^{x+y}

    Show that every complex number is the square of a complex number.
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  2. #2
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    Quote Originally Posted by terr13 View Post
    For a real number b>0 and a real number x, define b^x, and show b^xb^y = b^{x+y}

    Show that every complex number is the square of a complex number.
    It's hard to believe you are serious- or that whoever gave you those problems is serious. Both of these are very deep proofs. For the first one it is probably best to start by proving it for x and y positive integers and work your way up through the various number systems (integers, rational numbers, real numbers).

    The second is a special case of the "fundamental theorem of algebra" but perhaps not as hard as I thought at first. Given any complex number a+ bi, write (x+ yi)^2= a+ bi. Multiply it out, equate real part to real part, and solve the two real number equations for x and y.
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