Why is i^-3 the same as i^3?

Question

I don't understand why i^-3 wouldn't be 1/i^3
or 1/-i...can someone explain this to me?

Answer 1

i^-3 = 1/i^3 = i/i^4 = i/1 = i
i^3 = i(i²) = i(-1) = -i

they're NOT the same.

Answer 2

(i^-3) = (1 / -i) = i = (-i^3) ≠ (i^3)

See below for explanation:
========================

i^(-3)
= 1 / (i^3)
= 1 / -i

Now multiply the numerator and denominator by "i" to get "i" out of the denominator (kind of like how when you work with radicals, most teachers will ask you to express your final answer with no roots in the denominator):

1 / -i
= (1 * i) / (-i * i)
= i / -(i²)
= i / -(-1)
= i / 1
= i
= i(-1)(-1)
= i(i²)(-1)
= i³(-1)
= -i³    (not +i³)

In summary, (i^-3) = (1 / -i) = i = (-i^3) ≠ (i^3)

Answer 3

i^(-3) = 1/ i^3 = 1/ (-i) = i
i^3=-i

--> i^(-3)= - i^3 not i^3

Saludos.

Answer 4

Because when you square a negative number the result is the same as when you square the same number if it is positive.

Answer 5

actually, it will be...

i^-3 = 1/i^3

maybe its because of the i

note that:

i = √-1
i² = -1
i^3 = -i
i^4 = 1