A rational number in its lowest form has denominator $2^3 \times 5$. How many decimal places will its decimal expansion have? Explain your answer.

Denominator $q = 2^3 \times 5 = 40$.

Make it a power of 10, multiply numerator and denominator by $5^2 = 25$:

$$\frac{p}{2^3 \times 5} = \frac{25p}{2^3 \times 5 \times 5^2} = \frac{25p}{2^3 \times 5^3} = \frac{25p}{1000}$$

The denominator becomes $10^3$, so the decimal has 3 decimal places.

In general, for $q = 2^a \times 5^b$, the number of decimal places is $\max(a, b)$.

Here $a = 3, b = 1$, so $\max = 3$.