Express 5.165 (recurring) as a fraction in its simplest form.

Let $m$ be the repeating decimal:

$$m = 5.165165165... \quad (1)$$

Since there are three repeating digits (165), multiply equation $(1)$ by 1000:

$$1000m = 5165.165165... \quad (2)$$

Subtract equation $(1)$ from equation $(2)$:

$$1000m - m = 5165.165165... - 5.165165165...$$ $$999m = 5160$$

Divide both sides by 999:

$$m = \frac{5160}{999}$$

Both the numerator (5160) and the denominator (999) are divisible by 3.

$$5160 \div 3 = 1720$$ $$999 \div 3 = 333$$

So, the simplified improper fraction is:

$$m = \frac{1720}{333}$$

Divide 1720 by 333:

$$1720 = 5 \times 333 + 55$$

So, the mixed number is:

$$m = 5 \frac{55}{333}$$

The final answer is $\boxed{5 \frac