help with this math problem?

Question

how can i express this in simplest from : i^16+i^6-2i^5+i^13

please explain, i appreciate everyone that helps

Answer 1

Use the following to help solve this problem:

i ^1 = i
i ^2 = -1
i ^3 = -i
i ^4 = 1

This pattern then repeats:

i ^5 = i^4 * i = 1 * i = i
i ^6 = i^4 * i^2 = 1 * -1 = -1

etc.

So to solve any i ^n, just look at the remainder of n/4:

remainder
of n/4 ....... value of i ^n
- - - - - - - - - - - - - - - - - - -
0 ....................... 1
1 ........................ i
2 ....................... -1
3 ........................ -i

So now let's tackle your specific problem:

i ^16 = 1 (since n=16, n/4 = 4 with remainder of 0)
i ^6 = -1 (since n=6, n/4 = 1 with remainder of 2)
i ^5 = i (since n=5, n/5 = 1 with remainder of 1)
i ^13 = i (since n=13, n/4 = 3 with remainder of 1)

So, i^16 + i^6 - 2i^5 + i^13 =
1 + (-1) - 2(i) + i =
-i

Hope that helped!

~ ♥ ~

PS -- The "2-i" answers are incorrect. Those answerers just made a tiny typo and took 1 - (-1) as the first two terms, rather than 1 + (-1). Just wanted to let you know, so that you don't put the wrong answer down on your homework.

Answer 2

i goes in cycles of 4.
i^1 = i
i^2 = -1
i^3 = -i
i^4 = 1
i^5 = i etc..

So i^16 = 1, i^6 = -1, i^5 = i, and i ^13 = i.

So we have 1 - 1 - 2i + i = i. So the simplest form is -i.

Answer 3

keep in mind that:

i^0 = 1
i^1 = i
i^2 = -1
i^3 = -i

now for each "i" in your equation, divide the exponent by four and note the remainder..

i^16: 16 goes into four with a remainder of 0.. so that expression simplifies to what i^0 is which is 1
i^6: 6 goes into four with a remainder of 2.. so that expression simplifies to what i^2 is which is -1
ignore the -2 in the next expression for a while and take i^5: 5 goes into four with a remainder of 1.. so that expression simplifies to what i^1 is which is i
i^13: 13 goes into four with a remainder of 1.. so that expression simplifies to what i^1 is which is i

now we have: 1 -1 -2i +i
(basically substitution)

1-1 is 0..
-2i +i is -i

so the final answer is -i
now treat yourself to some candy!! we're done!! woot woot!!

Answer 4

It will be -i. Using the fact that i^2 = -i.
so i^16=1, i^6=-1, 2i^5=2i, i^13=i

Answer 5

Look for the largest power: it is 5 from the term -2i^5, then factor it out:
i^5[ i^11 + i -2 + i^8]

Answer 6

i^5(i^11 + i - 2 + i^8)

Answer 7

i^2 = -1
i^4 = 1
i^16+i^6-2i^5+i^13
(i^4)(i^4)(i^4)(i^4) - (i^4)(i^2) - 2(i^4)i + (i^4)(i^4)(i^4)i
1 - (-1) - 2i + i
2-i

ANSWER
2 - i

Answer 8

Oh God Im lost!