How do you simplify #i^656#?

1 Answer
Jan 13, 2016

#i^656 = 1#

Explanation:

We will use the following:

  • #i^2 = -1 => (i^2)^2 = (-1)^2 = 1#

  • #(x^a)^b = x^(ab)#

As #656 = 4*164# this means

#i^656 = i^(4*164) = (i^4)^164 = 1^164 = 1#

In general, it is easy to evaluate #i^n# by "pulling out" the greatest multiple of #4# possible from the exponent. Either the exponent will be a multiple of #4#, as above, or we will be left with #i#, #i^2#, or #i^3#. For example:

#i^23 = i^20i^3 = i^(4*5)i^3 = (i^4)^5i^3 = i^3 = i^2i = (-1)i = -i#