Answer :
To determine which of the given numbers are perfect squares, we need to identify numbers that can be expressed as the square of an integer. Let's analyze each number step-by-step:
1. 11:
- To check if 11 is a perfect square, we look for an integer [tex]\( n \)[/tex] such that [tex]\( n^2 = 11 \)[/tex].
- The square root of 11 ([tex]\( \sqrt{11} \)[/tex]) is approximately 3.316, which is not an integer.
- Therefore, 11 is not a perfect square.
2. 12:
- To check if 12 is a perfect square, we look for an integer [tex]\( n \)[/tex] such that [tex]\( n^2 = 12 \)[/tex].
- The square root of 12 ([tex]\( \sqrt{12} \)[/tex]) is approximately 3.464, which is not an integer.
- Therefore, 12 is not a perfect square.
3. 16:
- To check if 16 is a perfect square, we look for an integer [tex]\( n \)[/tex] such that [tex]\( n^2 = 16 \)[/tex].
- The square root of 16 ([tex]\( \sqrt{16} \)[/tex]) is exactly 4, which is an integer.
- Therefore, 16 is a perfect square.
4. 32:
- To check if 32 is a perfect square, we look for an integer [tex]\( n \)[/tex] such that [tex]\( n^2 = 32 \)[/tex].
- The square root of 32 ([tex]\( \sqrt{32} \)[/tex]) is approximately 5.657, which is not an integer.
- Therefore, 32 is not a perfect square.
5. 36:
- To check if 36 is a perfect square, we look for an integer [tex]\( n \)[/tex] such that [tex]\( n^2 = 36 \)[/tex].
- The square root of 36 ([tex]\( \sqrt{36} \)[/tex]) is exactly 6, which is an integer.
- Therefore, 36 is a perfect square.
After examining each number, we find that the perfect squares among the given numbers are:
[tex]\[ 16 \text{ and } 36 \][/tex]
1. 11:
- To check if 11 is a perfect square, we look for an integer [tex]\( n \)[/tex] such that [tex]\( n^2 = 11 \)[/tex].
- The square root of 11 ([tex]\( \sqrt{11} \)[/tex]) is approximately 3.316, which is not an integer.
- Therefore, 11 is not a perfect square.
2. 12:
- To check if 12 is a perfect square, we look for an integer [tex]\( n \)[/tex] such that [tex]\( n^2 = 12 \)[/tex].
- The square root of 12 ([tex]\( \sqrt{12} \)[/tex]) is approximately 3.464, which is not an integer.
- Therefore, 12 is not a perfect square.
3. 16:
- To check if 16 is a perfect square, we look for an integer [tex]\( n \)[/tex] such that [tex]\( n^2 = 16 \)[/tex].
- The square root of 16 ([tex]\( \sqrt{16} \)[/tex]) is exactly 4, which is an integer.
- Therefore, 16 is a perfect square.
4. 32:
- To check if 32 is a perfect square, we look for an integer [tex]\( n \)[/tex] such that [tex]\( n^2 = 32 \)[/tex].
- The square root of 32 ([tex]\( \sqrt{32} \)[/tex]) is approximately 5.657, which is not an integer.
- Therefore, 32 is not a perfect square.
5. 36:
- To check if 36 is a perfect square, we look for an integer [tex]\( n \)[/tex] such that [tex]\( n^2 = 36 \)[/tex].
- The square root of 36 ([tex]\( \sqrt{36} \)[/tex]) is exactly 6, which is an integer.
- Therefore, 36 is a perfect square.
After examining each number, we find that the perfect squares among the given numbers are:
[tex]\[ 16 \text{ and } 36 \][/tex]
