document.write( "Question 70009: i90i91+i92i93 note real numbers are exponents that add up to i to the 366 power \n" ); document.write( "

Algebra.Com's Answer #49850 by bucky(2189) $\"\"$ $\"About$

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Recognize that by definition $\"i%5E2+=+-1\"$ and $\"i%5E1=i\"$ \r
\n" ); document.write( "\n" ); document.write( "From this we can develop the sequence:\r
\n" ); document.write( "\n" ); document.write( " $\"i+=+i\"$
\n" ); document.write( " $\"i%5E2+=+-1\"$
\n" ); document.write( " $\"i%5E3+=+i%5E2+%2A+i+=+%28-1%29%2Ai+=+-i\"$
\n" ); document.write( " $\"i%5E4+=+i%5E2%2Ai%5E2+=+%28-1%29%2A%28-1%29+=+1\"$
\n" ); document.write( " $\"i%5E5+=+i%5E4%2Ai+=+1%2Ai+=+i\"$
\n" ); document.write( " $\"i%5E6+=+i%5E3%2Ai%5E3+=+%28-i%29%2A%28-i%29+=+i%5E2+=+-1\"$
\n" ); document.write( " $\"i%5E7+=+i%5E6%2Ai%5E1+=+%28-1%29%2Ai+=+-i\"$
\n" ); document.write( " $\"i%5E8+=+i%5E4%2Ai%5E4+=+1%2A1+=+1\"$
\n" ); document.write( " $\"i%5E9+=+i%5E8%2Ai%5E1+=+1%2Ai+=+i\"$ \r
\n" ); document.write( "\n" ); document.write( "Note that this series is i, -1, -i, 1, i, -1, -i, 1, i, ....\r
\n" ); document.write( "\n" ); document.write( "We can use this table along with some rules of exponents to simplify the terms in the problem.\r
\n" ); document.write( "\n" ); document.write( "Let's try to simplify $\"i%5E90\"$ . We use one of the laws of exponents to rewrite
\n" ); document.write( "the this as $\"i%5E90+=+%28i%5E9%29%5E10\"$ . From the above table you can see that $\"i%5E9+=+i\"$
\n" ); document.write( "Substitute this to get $\"i%5E90+=+%28i%29%5E10\"$ . We can further use a law of exponents
\n" ); document.write( "to factor this term into $\"i%5E10+=+i%5E5%2Ai%5E5\"$ . But $\"i%5E5+=+i\"$ so:
\n" ); document.write( " $\"i%5E5%2Ai%5E5=i%2Ai=i%5E2+=+-1\"$
\n" ); document.write( "So $\"i%5E90=-1\"$ .\r
\n" ); document.write( "\n" ); document.write( "We can use this result to find $\"i%5E91\"$ . Note that $\"i%5E91+=+i%5E90%2Ai%5E1\"$ . Substitute
\n" ); document.write( "-1 for $\"i%5E90\"$ to get $\"i%5E91+=+%28-1%29%2Ai+=+-i\"$ .\r
\n" ); document.write( "\n" ); document.write( "Then use this result and procedure to find that:
\n" ); document.write( " $\"i%5E92+=+i%5E91%2Ai%5E1+=%28-i%29%2Ai+=+-i%5E2+=+-%28-1%29+=+1\"$ \r
\n" ); document.write( "\n" ); document.write( "Repeat this process to find that:
\n" ); document.write( " $\"i%5E93+=+i%5E92%2Ai%5E1+=+1%2Ai+=+i\"$ \r
\n" ); document.write( "\n" ); document.write( "Substitute these four results into the original problem:
\n" ); document.write( " $\"%28i%5E90%29%2A%28i%5E91%29+%2B+%28i%5E92%29%2A%28i%5E93%29\"$ \r
\n" ); document.write( "\n" ); document.write( "The substitution results in:\r
\n" ); document.write( "\n" ); document.write( " $\"%28-1%29%2A%28-i%29+%2B+%281%29%2A%28i%29+=+i+%2B+i+=+2i\"$ \r
\n" ); document.write( "\n" ); document.write( "The answer to your problem is 2i.\r
\n" ); document.write( "\n" ); document.write( "Hope this helps.
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