Question

How do you simplify `8^(2/3)`?

Answer 1

`=4`

Explanation:

`8^(2/3)` can be written as
`root3(8^2`
`=root3(64)`
`=root3((4)(4)(4)`
`=4`

Answer 2

`4`

Explanation:

One of the laws of indices states: `x^(m/n) = rootn((x^m)) = (rootn(x))^m`

Therefore `8^(2/3)` can be written as `root3((8^2)) or (root3(8))^2`

I prefer to use the second form because it uses smaller numbers - the root is found first and then that is squared.

`root3(8) = 2 and 2^2 =4`

So: `(color(blue)(root3(8)))^2 = color(blue)(2)^2 = 4`

Consider a question such as `32^(3/5)`

`root5(color(blue)((32^3)))` would mean finding `32^3` first ... ouch!

`(color(blue)(root5 (32)))^3` would mean finding `root5(32)` first. That is `color(blue)(2)`

`(color(blue)(root5 (32)))^3 = color(blue)(2)^3 = 8`