document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209625>Question 1209625</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Express<BR>\n" );
document.write( "\cfrac{1}{1 - \cfrac{1}{2 + \cfrac{1}{i}}}<BR>\n" );
document.write( "in the form a+bi, where a and b are real numbers.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849735 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849735\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to simplify the complex fraction:\r<BR>\n" );
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document.write( "1. **Simplify the innermost fraction:**\r<BR>\n" );
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document.write( "   1/i = (1/i) * (i/i) = i/i² = i/(-1) = -i\r<BR>\n" );
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document.write( "2. **Substitute back:**\r<BR>\n" );
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document.write( "   The expression becomes:\r<BR>\n" );
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document.write( "   1 / (1 - 1 / (2 - i))\r<BR>\n" );
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document.write( "3. **Simplify the fraction in the denominator:**\r<BR>\n" );
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document.write( "   1 / (2 - i) = (1 / (2 - i)) * ((2 + i) / (2 + i)) = (2 + i) / (4 - i²) = (2 + i) / (4 + 1) = (2 + i) / 5 = 2/5 + (1/5)i\r<BR>\n" );
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document.write( "4. **Substitute back again:**\r<BR>\n" );
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document.write( "   1 / (1 - (2/5 + (1/5)i))\r<BR>\n" );
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document.write( "5. **Simplify the denominator:**\r<BR>\n" );
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document.write( "   1 - (2/5 + (1/5)i) = 1 - 2/5 - (1/5)i = 3/5 - (1/5)i\r<BR>\n" );
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document.write( "6. **Rewrite the expression:**\r<BR>\n" );
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document.write( "   1 / (3/5 - (1/5)i)\r<BR>\n" );
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document.write( "7. **Multiply by the conjugate of the denominator:**\r<BR>\n" );
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document.write( "   (1 / (3/5 - (1/5)i)) * ((3/5 + (1/5)i) / (3/5 + (1/5)i)) = (3/5 + (1/5)i) / ((3/5)² - (1/5)²i²) = (3/5 + (1/5)i) / (9/25 + 1/25) = (3/5 + (1/5)i) / (10/25) = (3/5 + (1/5)i) / (2/5)\r<BR>\n" );
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document.write( "8. **Simplify:**\r<BR>\n" );
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document.write( "   (3/5 + (1/5)i) * (5/2) = 3/2 + (1/2)i\r<BR>\n" );
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document.write( "Therefore, the expression in the form a + bi is 3/2 + (1/2)i.<BR>\n" );
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