document.write( "Question 1118160: Use De Moivres Theorem to show which integral powers of $\"+%28-1%2Bi%29%2F%28sqrt+%282%29%29+\"$ are real, and which are imaginary. \n" ); document.write( "

Algebra.Com's Answer #733428 by KMST(5328) $\"\"$ $\"About$

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Integral powers of $\"+%28-1%2Bi%29%2F%28sqrt+%282%29%29\"$ are real
\n" ); document.write( "if the integer exponent is a multiple of 4 ( such as 0,4,8,..., or -4,-8, -12,...).
\n" ); document.write( "They are imaginary if the integer exponent is a multiple of 2, but not a multiple of 4
\n" ); document.write( "(such as 2,6,10,..., or -2,-6,-10,...).
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\n" ); document.write( "If $\"p\"$ is an integer,
\n" ); document.write( " $\"+%28%28-1%2Bi%29%2F%28sqrt+%282%29%29%29%5Ep\"$ $\"%22=%22\"$ $\"%22%5B%22\"$ $\"cos%28pi%2F4%29%2Bi%2Asin%28pi%2F4%29\"$ $\"%22%5D%22%5Ep\"$ $\"%22=%22\"$ $\"cos%28p%2Api%2F4%29%2Bi%2Asin%28p%2Api%2F4%29\"$
\n" ); document.write( "will be real if $\"sin%28p%2Api%2F4%29=0\"$ ,
\n" ); document.write( "and will be imaginary if $\"cos%28p%2Api%2F4%29=0\"$ .
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\n" ); document.write( " $\"sin%28p%2Api%2F4%29=0\"$ if and only if $\"p%2Api%2F4=k%2Api\"$ for some integer $\"k\"$ ,
\n" ); document.write( "That means That the integral power $\"+%28%28-1%2Bi%29%2F%28sqrt+%282%29%29%29%5Ep\"$ is real only if and only if
\n" ); document.write( " $\"p\"$ is a multiple of 4.
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\n" ); document.write( " $\"cos%28p%2Api%2F4%29=0\"$ if and only if $\"p%2Api%2F4=k%2Api%2Bpi%2F2=%28%282k%2B1%29%2F2%29%2Api\"$ for some integer $\"k\"$ ,
\n" ); document.write( "That means That the integral power $\"+%28%28-1%2Bi%29%2F%28sqrt+%282%29%29%29%5Ep\"$ is imaginary if and only if, for some integer $\"k\"$ ,
\n" ); document.write( " $\"p%2F4=%282k%2B1%29%2F2\"$ <--> $\"p=4k%2B2\"$ ,
\n" ); document.write( "which means $\"p\"$ is even, but not a multiple of 4. \n" ); document.write( "

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