Question

How do you simplify `i^10+i^2`?

Answer 1

The expression simplifies to `-2`.

Explanation:

`i^10 + i^2`

Use the exponent rule `(x^a)^b= x^(ab)` and change the `i^10` term to an `i^2` term by dividing the exponent `color(blue)(10)` by `color(red)2=color(magenta)5`.

`i^color(blue)(10)=(i^color(red)2)^color(magenta)5`

`(i^2)^5+i^2`

Recall that `i^2=-1`

`(-1)^5 +(-1)`

`-1` raised to an odd power is equal to `-1` (and btw, `-1` raised to an even power is equal to `1`)

`(-1) +(-1)=-2`