Byron S. answered • 11/20/14

Tutor

5.0
(44)
Math and Science Tutor with an Engineering Background

Hi Sarah,

The key to this problem is understanding the definition of the number

*i*and what happens when you raise it to any power.By definition,

*i*

^{2}= -1

Using the product property of exponents,

*i*

^{3}=

*i**

*i*

^{2}= -

*i*

*i*

^{4}=

*i*

^{2}*

*i*

^{2}= (-1)(-1) = 1

For any power greater than 4, you can factor out

*i*^{4}, and this factor always reduces to 1.*i*

^{5}=

*i**

*i*

^{4}=

*i*

*i*

^{6}=

*i*

^{2}*

*i*

^{4}=

*i*

^{2}= -1

*i*

^{7}=

*i*

^{3}*

*i*

^{4}=

*i*

^{3}= -

*i*

*i*

^{8}=

*i*

^{4}*

*i*

^{4}= 1*1 = 1

We can simplify this problem by determining what the remainder of the exponent is after factoring out (dividing) all possible 4s from it.

(

*i*^{7})^{777}=

*i*^{5439}Multiples of 100 are always divisible by 4 (since 100 = 4*25), so we can pull them out

=

*i*^{5400}**i*^{39}=

*i*^{39}Now you need to find the largest multiple of 4 that is less than your exponent. In this case 4*9=36

=

*i*^{36}**i*^{3}=

*i*^{3}And from above,

=

**-***i*