The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form `p/q`, what can you say about the prime factors of q? (i)`43. 123456789` (ii) `0.120120012000120000` (iii) `43.overline(123456789)`

Explanation

To determine whether the given decimal expansions are rational or irrational, we will analyze each case step by step.

### Step 1: Analyze the first decimal expansion `43.123456789`
1. **Identify the type of decimal**: The decimal `43.123456789` has a finite number of digits after the decimal point (it stops at 9).
2. **Conclusion**: Since it is a terminating decimal, it is a rational number.

### Step 2: Analyze the second decimal expansion `0.120120012000120000`
1. **Identify the type of decimal**: The decimal `0.120120012000120000` shows a pattern where the number of zeros between the repeating digits is increasing (1, 2, 3, ...).
...