document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1026549>Question 1026549</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Could someone please check my work on the following paragraph proof?  This is to show why 11^1/n must equal n(sqrt)11 to uphold the properties of exponents, fill in the blanks to complete the paragraph proof.  The answers I filled in are between **  Suppose b^n=11.  Then b is the **nth** root of 11, which is written as **11^1/n**.  Now consider that 11=11^1 and that n/n=1.  By substitution, 11^1=11^n/n and by the **Power of a Power** property of exponents, 11^n/n=11^(1/n*n)=**(11^1/n)^n**.  By the transitive property, since b^n=11 and 11=**(11^1/n)^n**, we know that b^n=**(11^1/n)^n)**.  Therefore, b=11^1/n and as previously shown b=**n(sqrt)11**.  Thus, **n(sqrt)11=11^1/n**  Sorry it's so long, but I would appreciate any help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #641809 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=641809\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> At the very least I think you have to consider separating the proof by cases.\r<BR>\n" );
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document.write( "When n is even, it is trickier.  For example, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-root%284%2C11%29%29%5E4+=+%2811%5E%281%2F4%29%29%5E4 ALIGN=MIDDLE  ALT=\"%28-root%284%2C11%29%29%5E4+=+%2811%5E%281%2F4%29%29%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > doesn't mean  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-root%284%2C11%29+=+11%5E%281%2F4%29 ALIGN=MIDDLE  ALT=\"-root%284%2C11%29+=+11%5E%281%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "For n odd, it is automatic.\r<BR>\n" );
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document.write( "With that said, we have to impose the restriction that the root itself be a real number for the argument to hold. If complex roots are allowed then the notion of \"roots\" itself is not a single, particular number, but a class of (complex) numbers, and as such cannot be equal to just one number. </SPAN>\n" );
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