Answer:
1) multiplicative inverse of i = -i
2) Multiplicative inverse of i^2 = -1
3) Multiplicative inverse of i^3 = i
4) Multiplicative inverse of i^4 = 1
Step-by-step explanation:
We have to find multiplicative inverse of each of the following.
1) i
The multiplicative inverse is 1/i
if i is in the denominator we find their conjugate
[tex]=1/i * i/i\\=i/i^2\\=We\,\, know\,\, that\,\, i^2 = -1\\=i/(-1)\\= -i[/tex]
So, multiplicative inverse of i = -i
2) i^2
The multiplicative inverse is 1/i^2
We know that i^2 = -1
1/-1 = -1
so, Multiplicative inverse of i^2 = -1
3) i^3
The multiplicative inverse is 1/i^3
We know that i^2 = -1
and i^3 = i.i^2
[tex]1/i^3\\=1/i.i^2 \\=1/i(-1)\\=-1/i * i/i\\=-i/i^2\\= -i/-1\\= i[/tex]
so, Multiplicative inverse of i^3 = i
4) i^4
The multiplicative inverse is 1/i^4
We know that i^2 = -1
and i^4 = i^2.i^2
[tex]=1/i^2.i^2\\=1/(-1)(-1)\\=1/1\\=1[/tex]
so, Multiplicative inverse of i^4 = 1
