arithmetic - Repeating Decimal in different base

I've come across the following question.

Find $0.\overline{204}_6$ as a base ten fraction.

I understand that is the question asked the repeating decimal in base $10$, I would then say that:

$$x = 0.\overline{204}_{10}$$

$$1000x = 204.\overline{204}_{10}$$

$$999x = 204$$

$$x = \frac{204}{999}$$

Therefore, I can perform a similar task by doing this:

$$x = 0.\overline{204}_{6}$$

$$216x = 204.\overline{204}_{6}$$

$$216x = 204$$

$$x = \frac{204}{216} = \frac{17}{18}$$

Is the following procedure correct? Since initially we had to multiply by $10^3$ in base $10$, I would then assume that you would have to multiply by $6^3$, or $216$ in base $6$.