document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=25458>Question 25458</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (a)Find the (multiplicative) inverse of the complex number (3-i)^2 in the form a+ib, a,b belongs to the reals(R).\r<BR>\n" );
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document.write( "(b) solve the equation (((1+i)^5)x)= ((1-i)^3) in the ring C of complex numbers (also as number of the form a+ib, a,b belong to the reals(R). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #14435 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=14435\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> (a)Find the (multiplicative) inverse of the complex number (3-i)^2 in the form a+ib, a,b belongs to the reals(R).<BR>\n" );
document.write( "LET A+ib=1/(3-i)^2<BR>\n" );
document.write( "(A+iB)(3-i)^2=1<BR>\n" );
document.write( "LHS=(A+iB)((9+i^2-6i)=(A+iB)(9-1-6i)=(A+iB)(8-6i)<BR>\n" );
document.write( "=8A-6Ai+8Bi-6Bi^2=(8A+6B)+i(8B-6A)=RHS=1=1+0*i...HENCE EQUATING REAL AND IMAGINARY PARTS,WE GET <BR>\n" );
document.write( "8A+6B=1.........I<BR>\n" );
document.write( "8B-6A=0...OR.....4B-3A=0.............II<BR>\n" );
document.write( "2*EQN.I-3*EQN.II GIVES US <BR>\n" );
document.write( "16A+12B-2-12B+9A=0<BR>\n" );
document.write( "25A=2<BR>\n" );
document.write( "A=2/25<BR>\n" );
document.write( "B=3A/4...=3*2/(25*4)=3/50<BR>\n" );
document.write( "HENCE 1/(3-i)^2 = (2/25)+(3/50)*i\r<BR>\n" );
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document.write( "(b) solve the equation (((1+i)^5)x)= ((1-i)^3) in the ring C of complex numbers (also as number of the form a+ib, a,b belong to the reals(R).<BR>\n" );
document.write( "(((1+i)^5)x)= ((1-i)^3) <BR>\n" );
document.write( "X=(1-i)^3/(1+i)^5=((1-i)^3)((1-i)^5)/((1-i)^5(1+i)^5)=<BR>\n" );
document.write( "(1-i)^8/((1+i)(1-i))^5=(1-i)^8/(1-i^2)^5=(1-i)^8/32<BR>\n" );
document.write( "NOW WE HAVE <BR>\n" );
document.write( "(1-i)^2=1+i^2-2i=1-1-2i=-2i<BR>\n" );
document.write( "(1-i)^8={(1-i)^2}^4=(-2i)^4=(-2^4)(i^2)^2=16(-1)^2=16<BR>\n" );
document.write( "HENCE WE GET 16/32=1/2 AS THE ANSWER FOR X<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );