Your way looks fine, except for your final result:

$$5 \cdot 8^2 + 4 \cdot 8^1 + 7 \cdot 8^0 = 5 \cdot 64 + 4 \cdot 8 + 7 \cdot 1 = 320+32+7=359$$

A second way is Horner's method (only for decimal-system) for using in calculation:

You also start with $359=5 \cdot 8^2 + 4 \cdot 8^1 + 7 \cdot 8^0$. Thats a sum of 3 products (left factor is your number, right factor is a multiple of 8). Re-arrange:

$$1\cdot 7 + 4 \cdot 8 + 5 \cdot 8^2=359$$

$$7+8\cdot (4+8\cdot 5)=359$$

Now you can use a calculator:

     Input              Output
        5                5
   [x]  8  [+]  4  [=]   44
   [x]  8  [+]  7  [=]   359

Hope this helps.