Introduction:
In order to determine the smallest length that can be measured using two rods of lengths 9 feet and 6 feet, we need to consider the concept of finding the greatest common divisor (GCD) of the two lengths.

GCD:
The GCD is the largest number that divides both of the given lengths without leaving a remainder. It helps us find the common unit length that can be used to measure various lengths using the given rods.

Calculating GCD:
To find the GCD of 9 feet and 6 feet, we can use the Euclidean algorithm.

Step 1: Divide the larger length (9 feet) by the smaller length (6 feet). The quotient is 1 and the remainder is 3 feet.
Step 2: Now, divide the smaller length (6 feet) by the remainder (3 feet). The quotient is 2 and there is no remainder.
Step 3: Since we have obtained a remainder of 0, we stop here. The GCD of 9 feet and 6 feet is the last non-zero remainder, which is 3 feet.

Explanation:
The GCD of 9 feet and 6 feet is 3 feet. This means that the rods can be used to measure lengths that are multiples of 3 feet. By aligning the rods end to end, we can measure lengths of 3 feet, 6 feet, 9 feet, and so on.

Smallest Length:
To find the smallest length that can be measured using either of the rods, we need to find the LCM (Least Common Multiple) of the two lengths. The LCM is the smallest number that is divisible by both 9 feet and 6 feet.

Calculating LCM:
To find the LCM of 9 feet and 6 feet, we can use the formula:
LCM = (9 feet * 6 feet) / GCD

Substituting the values, we have:
LCM = (9 * 6) / 3 = 18 feet

Therefore, the smallest length that can be measured using either of the rods is 18 feet.

Conclusion:
The GCD of 9 feet and 6 feet is 3 feet, which represents the common unit length that can be measured using the rods. The LCM of 9 feet and 6 feet is 18 feet, which is the smallest length that can be measured using either of the rods.