document.write( "Question 420878: (cos45+isin45)^8 \n" ); document.write( "

Algebra.Com's Answer #293980 by jsmallt9(3758) $\"\"$ $\"About$

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The easy way to raise a complex number to a power is DeMoivre's Theorem:
\n" ); document.write( "For any complex number
\n" ); document.write( " $\"z+=+r%28cos%28x%29+%2B+i%2Asin%28x%29%29\"$
\n" ); document.write( "then
\n" ); document.write( " $\"z%5En+=+r%5En%2A%28cos%28n%2Ax%29+%2B+i%2Asin%28n%2Ax%29%29\"$

\n" ); document.write( "For your complex number, $\"cos%2845%29%2Bi%2Asin%2845%29\"$ , there is no visible r. So the r must be a 1. Writing your expression with a visible r we get:
\n" ); document.write( " $\"%281%28cos%2845%29+%2B+i%2Asin%2845%29%29%29%5E8\"$
\n" ); document.write( "Now we can use DeMoivre's Theorem to rewrite your expression:
\n" ); document.write( "1^8(cos(8*45) + i*sin(8*45))
\n" ); document.write( "which simplifies as follows:
\n" ); document.write( "1(cos(360) + i*sin(360))
\n" ); document.write( "1(1 + i*0)
\n" ); document.write( "1(1)
\n" ); document.write( "1
\n" ); document.write( "So (cos45+isin45)^8 = 1 \n" ); document.write( "

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