Complete the pattern: \[ \begin{array}{l}\square \div 10=7.764 \\ \square 1,000=0.07764 \\ \square 10,000=0.007764 \\ \square 100,000=0.0007764\end{array} \]

To complete the pattern, we need to observe the relationships among the numbers. 1. The first equation shows the square root of a number divided by 10 being equal to 7.764, suggesting that the missing number is 77.64 (since \( 77.64 \div 10 = 7.764 \)). 2. The second equation shows that multiplying 77.64 by 1,000 gives 0.07764, which can be seen as moving the decimal point three places to the left. 3. In the third equation, when 77.64 is multiplied by 10,000, it results in 0.007764; that’s adding another place to the left in the decimal. 4. In the fourth equation, multiplying by 100,000 gives 0.0007764, pushing the decimal once more. Continuing this pattern, each subsequent operation involves moving the decimal point further left as you multiply by higher powers of 10. Hence, the missing entries will involve consistently reducing the value by powers of ten with each step. In sum, the missing number in the sequence is 77.64, and to represent the pattern in each case: - 77.64 (as the first missing number) - 0.07764 (as the second) - 0.007764 (as the third) - 0.0007764 (as the fourth). This illustrates a clear trend of decreasing by a factor of ten with each operation, maintaining the flow of the sequence beautifully.