Rationale
√(45) is between 6 and 7.
The square root of 45 is approximately 6.7, which places it between the whole numbers 6 and 7. This can be confirmed by recognizing that 6² equals 36 and 7² equals 49, indicating that 45 lies between these two perfect squares.
A) 4 and 5
The squares of 4 and 5 are 16 and 25, respectively. Since 45 is much larger than 25, this pair of whole numbers does not enclose the square root of 45, making this option incorrect.
B) 5 and 6
The squares of 5 and 6 are 25 and 36, respectively. Again, 45 exceeds 36, meaning this interval does not contain the square root of 45, and thus this choice is incorrect.
C) 6 and 7
This choice is correct because 6² equals 36 and 7² equals 49. As such, 45 falls between these two values, confirming that the square root of 45 is indeed between 6 and 7.
D) 14 and 15
The squares of 14 and 15 are 196 and 225, respectively. Since 45 is far less than both of these values, this option is incorrect, as it does not encompass the square root of 45.
E) 22 and 23
The squares of 22 and 23 are 484 and 529, respectively. Since 45 is considerably lower than 484, this option also does not include the square root of 45, making it incorrect.
Conclusion
The square root of 45 lies between the whole numbers 6 and 7, as verified by calculating the squares of these integers. None of the other options correctly encompass the value of √(45), which further confirms that 6 and 7 is the only valid choice. Understanding the relationship between squares and square roots is essential for determining such intervals accurately.
