Question

Match each power of i with its multiplicative inverse.

1) i
2) i^2
3) i^3
4) i^4

Each number has one of the following answers answers cannot be used more than once.
Answer 1) i
Answer 2) 1
Answer 3) -i
Answer 4) -1

Answer 1

1. 
   `i=(1)/(i)*(-i)/(-1)=-i`
2. 
   `i^2=(1)/(i^2)*(-i^2)/(-i^2)=-1`
3. 
   `i^3=(1)/(i^3)*(-i^3)/(-i^3)=i`
4. 
   `i^4=(1)/(i^4)*(-i^4)/(-i^4)=1`

Explanation:

To find the multiplicative inverse of a complex number, you do as follows:

The inverse is found by reciprocating the original complex number. (
`(1)/(complex number)`) Multiply the numerator and denominator of the reciprocal by conjugate of the denominator and simplify.

Hope this helps!!

Answer 2

By definition we have to:
In mathematics, the inverse multiplicative, reciprocal or inverse of a non-zero number x, is the number, denoted as 1/x or x -1, which multiplied by x gives 1.
We have then:

1) i
The multiplicative inverse is:

`(1)/(i)`
Rewriting we have:

`(1)/(i) (i)/(i)`

`(i)/(i^2)`

`(i)/(-1)`

`-i`
Answer 3) -i

2) i^2
The multiplicative inverse is:

`(1)/(i^2)`
Rewriting we have:

`(1)/(-1)`

`-1`
Answer 4) -1

3) i^3
The multiplicative inverse is:

`(1)/(i^3)`
Rewriting we have:

`(1)/(i*i^2)`

`(1)/(i^2)*(1)/(i)`

`(1)/(-1)*(1)/(i)*(i)/(i)`

`(-1)*(i)/(i^2)`

`(-1)* (i)/(-1)`

`i`
Answer 1) i

4) i^4
The multiplicative inverse is:

`(1)/(i^4)`
Rewriting we have:

`(1)/(i^2i^2)`

`(1)/((-1)(-1))`

`(1)/(1)`

`1`
Answer 2) 1