The order of operations:
1. Take care of absolute values as they occur left to right.
2. Take care of parentheses as they occur left to right.
3. Take care of exponents and roots as they occur left to right.
4. Take care of multiplications and divisions as they occur left 
   to right.
5. Take care of additions and subtractions as they occur left 
   to right.

Just do one replacement each time, and copy everything else over.
Don't try to skip steps until you have had more experience.  

3^2-|-18|/9+2^3

First replace the absolute value:

3^2-|-18|/9+2^3

Replace |-18| by 18

Copy everything else over:

3^2-18/9+2^3

Remove the red:

3^2-18/9+2^3

Replace the first exponent which occurs left to right:

3^2-18/9+2^3

Replace 3^2 by 9

Copy everything else over:

9-18/9+2^3

Remove the red

9-18/9+2^3

Replace the remaining exponent:

9-18/9+2^3

Replace 2^3 by 8

Copy everything else over:

9-18/9+8

Remove the red:

9-18/9+8

Replace the multiplications or divisions, whichever comes first.
[There are no multiplications, only the division]

9-18/9+8

Replace 18/9 by 2

Copy everything else over:

9-2+8

Remove the red:

9-2+8

Do the first addition or subtraction (whichever comes first)
The subtraction comes first left to right, so

9-2+8

Replace 9-2 by 7

Copy everything else over:

7+8

Remove the red:

7+8

That's 

15

Edwin