How can I expand 9^n (where n is a number between 0 and 1)?

Question

With 9^2=81, the 9^2 can be expanded to 9*9.
With 9^3=729, the 9^3 can be expanded to 9*9*9.

So they are very obvious and easy to understand.

But for something like this:
9^0.5
9^1/3
Although I can still understand the first one is taking the square root of 9, and the second one is taking the cube root of 9. I can't manually expand them easily and I have to use the calculator. Taking 9^0.7 as an example, I can neither treat as the sqaure root nor cube root of 9, so it's even more difficult to understand.

I want to expand them so that they are easier to understand, is this possible?

Thank you.

Answer 1

Let y = 9^x, where x is a real number
then
ln(y) = ln(9^x) = x ln(9)
or
y = 9^x = e^{ x ln9 }

Answer 2

Maybe this will help, maybe not.

9^0.7 is the same as 9^(7/10)

which is the 10th root of 9^7

which is also the 10th root of 9*9*9*9*9*9*9

or the 10th root of 3*3*3*3*3*3*3*3*3*3*3*3*3*3

Factor out 10 3s and you get
3 times the 10th root of 81

which is 3 * 1.0288
or 3.0864

Help any???
Rousey

Answer 3

Aren't you basically applying laws of exponents?