document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209757>Question 1209757</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If z is a complex number satisfying\r<BR>\n" );
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document.write( "z + \frac{1}{z} = \sqrt{2},\r<BR>\n" );
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document.write( "calculate\r<BR>\n" );
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document.write( "z^{10} + \frac{1}{z^{10}}. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850185 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=850185\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Rewrite the Given Equation:**\r<BR>\n" );
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document.write( "We are given:\r<BR>\n" );
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document.write( "z + 1/z = √2\r<BR>\n" );
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document.write( "Multiply both sides by z:\r<BR>\n" );
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document.write( "z² + 1 = √2z\r<BR>\n" );
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document.write( "Rearrange to form a quadratic equation:\r<BR>\n" );
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document.write( "z² - √2z + 1 = 0\r<BR>\n" );
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document.write( "**2. Solve for z:**\r<BR>\n" );
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document.write( "Using the quadratic formula:\r<BR>\n" );
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document.write( "z = [√2 ± √((√2)² - 4(1)(1))] / 2\r<BR>\n" );
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document.write( "z = [√2 ± √(2 - 4)] / 2\r<BR>\n" );
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document.write( "z = [√2 ± √(-2)] / 2\r<BR>\n" );
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document.write( "z = [√2 ± i√2] / 2\r<BR>\n" );
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document.write( "z = (√2 / 2) ± i(√2 / 2)\r<BR>\n" );
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document.write( "z = (1/√2) ± i(1/√2)\r<BR>\n" );
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document.write( "**3. Express z in Polar Form:**\r<BR>\n" );
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document.write( "We can express z in polar form as z = r(cos θ + i sin θ), where r is the magnitude and θ is the argument.\r<BR>\n" );
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document.write( "* Magnitude (r):<BR>\n" );
document.write( "    r = √[(1/√2)² + (1/√2)²] = √(1/2 + 1/2) = √1 = 1\r<BR>\n" );
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document.write( "* Argument (θ):<BR>\n" );
document.write( "    Since cos θ = 1/√2 and sin θ = ±1/√2, we have:<BR>\n" );
document.write( "    * If z = (1/√2) + i(1/√2), then θ = π/4.<BR>\n" );
document.write( "    * If z = (1/√2) - i(1/√2), then θ = -π/4.\r<BR>\n" );
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document.write( "Thus, we have:\r<BR>\n" );
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document.write( "* z = cos(π/4) + i sin(π/4) or<BR>\n" );
document.write( "* z = cos(-π/4) + i sin(-π/4)\r<BR>\n" );
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document.write( "**4. Use De Moivre's Theorem:**\r<BR>\n" );
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document.write( "De Moivre's Theorem states that for any complex number z = r(cos θ + i sin θ) and integer n:\r<BR>\n" );
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document.write( "z^n = r^n (cos(nθ) + i sin(nθ))\r<BR>\n" );
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document.write( "In our case, r = 1, so:\r<BR>\n" );
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document.write( "z^n = cos(nθ) + i sin(nθ)\r<BR>\n" );
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document.write( "**5. Calculate z^10:**\r<BR>\n" );
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document.write( "* If θ = π/4:<BR>\n" );
document.write( "    z^10 = cos(10π/4) + i sin(10π/4) = cos(5π/2) + i sin(5π/2) = cos(π/2) + i sin(π/2) = 0 + i(1) = i<BR>\n" );
document.write( "* If θ = -π/4:<BR>\n" );
document.write( "    z^10 = cos(-10π/4) + i sin(-10π/4) = cos(-5π/2) + i sin(-5π/2) = cos(-π/2) + i sin(-π/2) = 0 + i(-1) = -i\r<BR>\n" );
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document.write( "**6. Calculate 1/z^10:**\r<BR>\n" );
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document.write( "* If z^10 = i, then 1/z^10 = 1/i = -i.<BR>\n" );
document.write( "* If z^10 = -i, then 1/z^10 = 1/(-i) = i.\r<BR>\n" );
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document.write( "**7. Calculate z^10 + 1/z^10:**\r<BR>\n" );
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document.write( "* If z^10 = i and 1/z^10 = -i, then z^10 + 1/z^10 = i + (-i) = 0.<BR>\n" );
document.write( "* If z^10 = -i and 1/z^10 = i, then z^10 + 1/z^10 = -i + i = 0.\r<BR>\n" );
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document.write( "**Final Answer:**\r<BR>\n" );
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document.write( "z^10 + 1/z^10 = 0<BR>\n" );
document.write( " </SPAN>\n" );
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