Question

How do you simplify `sqrt(-2)^6`?

Answer 1

`(sqrt(-2))^6 = -8`

Explanation:

Note that if `a` is any number and `m, n` are positive integers then:

`(a^m)^n = overbrace((a^m)xx(a^m)xx...xx(a^m))^"n times"`

`color(white)((a^m)^n) = overbrace(overbrace((axxaxx...xxa))^"m times"xxoverbrace((axxaxx...xxa))^"m times"xx...xxoverbrace((axxaxx...xxa))^"m times")^"n times"`

`color(white)((a^m)^n) = overbrace(axxaxx...xxa)^"mn times"`

`color(white)((a^m)^n) = a^(mn)`

So in our example we find:

`(sqrt(-2))^6 = (sqrt(-2))^(2*3) = (sqrt(-2)^2)^3 = (-2)^3 = -8`

`color(white)()`
Footnote

I demonstrated `(a^m)^n = a^(mn)` above since this particular "rule" fails for negative or complex values of `a` if `m, n` are fractional.

For example:

`-1 = (-1)^1 = (-1)^(3/2*2/3) != ((-1)^(3/2))^(2/3) = (-i)^(2/3) = 1/2-sqrt(3)/2i`