Estimate the square root of 75 to the nearest whole number ?​
The Correct Answer and Explanation is :
To estimate the square root of 75 to the nearest whole number, we first identify two perfect squares between which 75 lies. The perfect squares closest to 75 are 64 (which is (8^2)) and 81 (which is (9^2)). Since 75 is greater than 64 and less than 81, we can conclude that the square root of 75 will be between 8 and 9.
To refine our estimate, we can calculate the squares of numbers between 8 and 9:
- (8.5^2 = 72.25)
- (8.6^2 = 73.96)
- (8.7^2 = 75.69)
From this, we see that (8.5^2) is 72.25, and (8.6^2) is 73.96, both of which are less than 75, while (8.7^2) equals 75.69, which is slightly greater than 75. This means that the square root of 75 is slightly less than 8.7.
Next, we can further narrow it down by checking (8.6):
- (8.6^2 = 73.96) (less than 75)
- (8.7^2 = 75.69) (greater than 75)
Since (75) is much closer to (73.96) than to (75.69), we can say that ( \sqrt{75} ) is approximately 8.6, which is less than halfway to 8.7. Thus, when rounding (8.6) to the nearest whole number, we arrive at 9.
To summarize, since (75) lies between the squares of 8 and 9, and more closely approximates 9, we conclude that the square root of 75 to the nearest whole number is 9.