Final answer:
The pattern under the square root is a product of 3 and consecutive perfect squares. The next number in the sequence is the square root of 3 multiplied by 5 squared, which is √75.
Step-by-step explanation:
To complete the pattern √3, √12, √27, √48, ......, we should look for a relationship between the numbers under the square root. Let's examine the pattern they form:
- The first number is √3, which we can also write as √(3×1).
- The second number is √12, which we can write as √(3×4).
- The third number is √27, which we can write as √(3×9).
- The fourth number is √48, which can be written as √(3×16).
Looking at this pattern, we notice that the multiplier of 3 is increasing by 3 each time (1, 4, 9, 16,...), which are perfect squares (1^2, 2^2, 3^2, 4^2,...).
The next number in the sequence would be 5^2 since we are incrementing the perfect squares by 1 sequentially. Therefore, the next number in the pattern is √(3×25) or √75.