Identify the two consecutive integers in which each square root lies between square root of 360
The Correct Answer and Explanation is:
To identify the two consecutive integers between which the square root of 360 lies, we can use the method of estimating square roots and checking the values of squares for integers.
Step 1: Find the square roots of integers near 360.
We know that:
- The square root of 324 is 18 because ( 18^2 = 324 ).
- The square root of 400 is 20 because ( 20^2 = 400 ).
So, ( \sqrt{360} ) lies between the square roots of 324 and 400, which are 18 and 20.
Step 2: Refine our estimate.
To get a more precise estimate, we can check the squares of numbers between 18 and 20:
- ( 19^2 = 361 ), which is very close to 360.
- ( 18^2 = 324 ), which is less than 360.
- ( 20^2 = 400 ), which is greater than 360.
Thus, the square root of 360 lies between 18 and 19.
Step 3: Conclusion.
The two consecutive integers between which ( \sqrt{360} ) lies are 18 and 19. We can conclude this because:
- ( 18^2 = 324 ), which is less than 360, and
- ( 19^2 = 361 ), which is very close to 360 but slightly larger.
Therefore, ( \sqrt{360} ) lies between 18 and 19.
This method of estimating square roots by finding nearby perfect squares is effective for approximating the square root of any number without using a calculator, and in this case, it tells us that the square root of 360 lies between the integers 18 and 19.