To determine if a number is a perfect square, we need to consider if there is an integer [tex]\( n \)[/tex] such that [tex]\( n^2 \)[/tex] equals the given number. Let's analyze each of the numbers provided:
a) 1296:
- To check if 1296 is a perfect square, we need to find some integer [tex]\( n \)[/tex] where [tex]\( n^2 = 1296 \)[/tex].
- By inspection, we find that 36 squared ([tex]\( 36 \times 36 \)[/tex]) equals 1296.
- Therefore, 1296 is a perfect square.
b) 1000:
- To determine if 1000 is a perfect square, we need to see if there exists an integer [tex]\( n \)[/tex] such that [tex]\( n^2 = 1000 \)[/tex].
- Calculations show that the square root of 1000 is not an integer, as it lies between 31 and 32 (since [tex]\( 31^2 = 961 \)[/tex] and [tex]\( 32^2 = 1024 \)[/tex]).
- Therefore, 1000 is not a perfect square.
c) 373758:
- We need to ascertain if 373758 can be represented as [tex]\( n^2 \)[/tex] for some integer [tex]\( n \)[/tex].
- The exact square root of 373758 is not a whole number, as it lies between 611 and 612 (since [tex]\( 611^2 = 373321 \)[/tex] and [tex]\( 612^2 = 374544 \)[/tex]).
- Thus, 373758 is not a perfect square.
d) 22034087:
- We examine if there exists [tex]\( n \)[/tex] such that [tex]\( n^2 = 22034087 \)[/tex].
- The square root of 22034087 does not result in an integer value. It lies between 4693 and 4694 (since [tex]\( 4693^2 = 22027249 \)[/tex] and [tex]\( 4694^2 = 22048676 \)[/tex]).
- Consequently, 22034087 is not a perfect square.
Summary:
- 1296: Perfect square.
- 1000: Not a perfect square.
- 373758: Not a perfect square.
- 22034087: Not a perfect square.
Thus, the numbers which are not perfect squares are:
1000, 373758, 22034087.
