document.write( "Question 12743: 4th root of negative one \n" ); document.write( "

Algebra.Com's Answer #6464 by khwang(438) $\"\"$ $\"About$

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Since -1 = cos pi + i sin pi
\n" ); document.write( " = cos (pi + 2k pi) + i sin (pi + 2 k pi)
\n" ); document.write( " = cos (2k+1) pi + i sin (2k+1) pi for integer k
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\n" ); document.write( " By De Moivre’s Theorem, if z = r(cos x + i sin x)
\n" ); document.write( " then $\"z%5En+\"$ = $\"r%5En\"$ (cos nx + i sin nx)
\n" ); document.write( " and $\"z%5E%281%2Fn%29+\"$ = $\"r%5E%281%2Fn%29\"$ (cos $\"x%2Fn\"$ + i sin $\"x%2Fn\"$ )
\n" ); document.write( " for integer n.\r
\n" ); document.write( "\n" ); document.write( " Hence, $\"%28-1%29%5E%281%2F4%29\"$ = cos ((2k+1)*pi /4) + i sin ( (2k+1)*pi /4) )
\n" ); document.write( " k= 0,1,2 or 3.\r
\n" ); document.write( "\n" ); document.write( " So, we have four 4th root of -1 as:
\n" ); document.write( " cos ( $\"pi%2F4\"$ ) + i sin ( $\"pi%2F4\"$ ), ( when k = 0) called the primitive 4th root of -1.
\n" ); document.write( " cos ( $\"3%2Api%2F4\"$ ) + i sin ( $\"3%2Api%2F4\"$ ), (when k = 1)
\n" ); document.write( " cos ( $\"5%2Api%2F4\"$ ) + i sin ( $\"5%2Api%2F4\"$ ), (when k = 2)
\n" ); document.write( " cos ( $\"7%2Api%2F4\"$ ) + i sin ( $\"7%2Api%2F4\"$ ), (when k = 3)\r
\n" ); document.write( "\n" ); document.write( " Note use pi/4 (radians) not degrees. (even we know it is 45 deg)
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\n" ); document.write( " This is very basic questions of complex numbers. Try to think about it.\r
\n" ); document.write( "\n" ); document.write( " Kenny\r
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