document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196798>Question 1196798</A>: <I><SPAN STYLE=\"word-break: break-all;\"> prove that √ 2/2 + √ 2/2i is a square root of i </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #829796 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=829796\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "If x is a square root of y, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+sqrt%28y%29 ALIGN=MIDDLE  ALT=\"x+=+sqrt%28y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which rearranges into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+y ALIGN=MIDDLE  ALT=\"x%5E2+=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Example: 7 is a square root of 49, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7=sqrt%2849%29 ALIGN=MIDDLE  ALT=\"7=sqrt%2849%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which turns into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%5E2+=+49 ALIGN=MIDDLE  ALT=\"7%5E2+=+49\"   DISABLED_style= \"max-width:90%; height:auto;\" > after squaring both sides.\r<BR>\n" );
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document.write( "The claim is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%2F2%2Bexpr%28sqrt%282%29%2F2%29i ALIGN=MIDDLE  ALT=\"sqrt%282%29%2F2%2Bexpr%28sqrt%282%29%2F2%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a square root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i ALIGN=MIDDLE  ALT=\"i\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i+=+sqrt%28-1%29 ALIGN=MIDDLE  ALT=\"i+=+sqrt%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "A useful rule of complex numbers is this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bbi%29%5E2+=+%28a%5E2-b%5E2%29+%2B+2abi ALIGN=MIDDLE  ALT=\"%28a%2Bbi%29%5E2+=+%28a%5E2-b%5E2%29+%2B+2abi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I'll leave it to the student to prove this claim.\r<BR>\n" );
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document.write( "In this case, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"a+=+sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"b+=+sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bbi%29%5E2+=+%28a%5E2-b%5E2%29+%2B+2abi ALIGN=MIDDLE  ALT=\"%28a%2Bbi%29%5E2+=+%28a%5E2-b%5E2%29+%2B+2abi\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%282%29%2F2%2Bexpr%28sqrt%282%29%2F2%29i%29%5E2+=+%28%28sqrt%282%29%2F2%29%5E2-%28sqrt%282%29%2F2%29%5E2%29+%2B+2%2A%28sqrt%282%29%2F2%29%2Aexpr%28sqrt%282%29%2F2%29i ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%282%29%2F2%2Bexpr%28sqrt%282%29%2F2%29i%29%5E2+=+0+%2B+1i ALIGN=MIDDLE  ALT=\"%28sqrt%282%29%2F2%2Bexpr%28sqrt%282%29%2F2%29i%29%5E2+=+0+%2B+1i\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%282%29%2F2%2Bexpr%28sqrt%282%29%2F2%29i%29%5E2+=+i ALIGN=MIDDLE  ALT=\"%28sqrt%282%29%2F2%2Bexpr%28sqrt%282%29%2F2%29i%29%5E2+=+i\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This verifies that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%282%29%2F2%2Bexpr%28sqrt%282%29%2F2%29i%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%282%29%2F2%2Bexpr%28sqrt%282%29%2F2%29i%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is indeed a square root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i ALIGN=MIDDLE  ALT=\"i\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In other words, one solution to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+i ALIGN=MIDDLE  ALT=\"x%5E2+=+i\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+sqrt%282%29%2F2%2Bexpr%28sqrt%282%29%2F2%29i ALIGN=MIDDLE  ALT=\"x+=+sqrt%282%29%2F2%2Bexpr%28sqrt%282%29%2F2%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The other solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-sqrt%282%29%2F2-expr%28sqrt%282%29%2F2%29i ALIGN=MIDDLE  ALT=\"x+=+-sqrt%282%29%2F2-expr%28sqrt%282%29%2F2%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" > which I'll leave to the reader to prove. <BR>\n" );
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