Longest tape to measure 84cm:
To determine the longest tape that can measure exactly 84cm, we need to find the common factor between 84 and the smallest unit of measurement on the tape. In this case, the smallest unit of measurement is usually 1cm.

- Find the factors of 84:
The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.

- Find the common factors between 84 and 1cm:
Since 84 is divisible by 1, we can use a tape with a smallest unit of measurement of 1cm to measure exactly 84cm.

Longest tape to measure 1m 38cm:
To determine the longest tape that can measure exactly 1m 38cm, we need to convert the measurement into a single unit, such as centimeters.

- Convert 1m 38cm into centimeters:
1 meter is equal to 100 centimeters, so 1m 38cm is equal to (1 * 100) + 38 = 138cm.

- Find the factors of 138:
The factors of 138 are 1, 2, 3, 6, 23, 46, 69, and 138.

- Find the common factors between 138 and 1cm:
Since 138 is divisible by 1, we can use a tape with a smallest unit of measurement of 1cm to measure exactly 1m 38cm or 138cm.

Longest tape to measure 12m:
To determine the longest tape that can measure exactly 12m, we need to convert the measurement into a single unit, such as centimeters.

- Convert 12m into centimeters:
1 meter is equal to 100 centimeters, so 12m is equal to 12 * 100 = 1200cm.

- Find the factors of 1200:
The factors of 1200 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 240, 300, 400, 600, and 1200.

- Find the common factors between 1200 and 1cm:
Since 1200 is divisible by 1, we can use a tape with a smallest unit of measurement of 1cm to measure exactly 12m or 1200cm.

Conclusion:
- The longest tape that can be used to measure exactly 84cm is a tape with a smallest unit of measurement of 1cm.
- The longest tape that can be used to measure exactly 1m 38cm is a tape with a smallest unit of measurement of 1cm.
- The longest tape that can be used to measure exactly 12m is a tape with a smallest unit of measurement of 1