document.write( "Question 26756: what is i to the 235th power \n" ); document.write( "

Algebra.Com's Answer #14557 by bmauger(101) $\"\"$ $\"About$

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i, the imaginary square root of -1, follows a pattern when you raise it to various powers.
\n" ); document.write( "By definition any number (even this imaginary one) to the 0th power is 1, so:
\n" ); document.write( " $\"i%5E0=1\"$
\n" ); document.write( "Since any number raised to 1 is itself, we start with:
\n" ); document.write( " $\"i%5E1=i\"$
\n" ); document.write( "by the definition of i, squaring it gives -1:
\n" ); document.write( " $\"i%5E2=-1\"$
\n" ); document.write( "taking it to the third power is the same as multiplying -1 (i squared) times i:
\n" ); document.write( " $\"i%5E3=-i\"$
\n" ); document.write( "and taking it to the forth power is the same as multiplying -1 (i squared) times -1 (i squared):
\n" ); document.write( " $\"i%5E4+=+i%5E2+%2A+i%5E2+=+%28-1%29%5E2+=+1\"$
\n" ); document.write( "If we multiply this by another i, i raised to the fifth, we start to see the pattern:
\n" ); document.write( " $\"i%5E5+=+i%5E4+%2A+i+=+1+%2A+i+=+i\"$
\n" ); document.write( "In fact, knowing that i^4=1 means that we can multiply as many i^4 as possible together and still get 1. Example:
\n" ); document.write( " $\"i%5E24+=+%28i%5E4%29%5E6+=+1%5E6+=+1\"$
\n" ); document.write( "Dividing 235 by 4, we get 58.75 so we have a total of 58 4's we can \"cancel\". Therefore, in your problem, i^235, we can rewrite 235 as 4*58 + 3. Using laws of exponents i to the 235th can be rewritten as:
\n" ); document.write( " $\"i%5E235+=+i%5E%284%2A58%2B3%29\"$
\n" ); document.write( " $\"i%5E235+=+%28i+%5E+4%29%5E58+%2A+i+%5E3\"$
\n" ); document.write( " $\"i%5E235+=+1%5E58+%2A+%28-i%29\"$
\n" ); document.write( "Answer:
\n" ); document.write( " $\"i%5E235+=+-i\"$ \r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

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