Let's solve the problem step by step:
1. Identify the number under the radical: [tex]\( \sqrt{36} \)[/tex].
2. Determine the factors of 36 that are perfect squares. The perfect squares less than or equal to 36 are:
- 1 ([tex]\(1^2\)[/tex])
- 4 ([tex]\(2^2\)[/tex])
- 9 ([tex]\(3^2\)[/tex])
- 16 ([tex]\(4^2\)[/tex]) — however, 16 is not a factor of 36.
- 25 ([tex]\(5^2\)[/tex]) — however, 25 is not a factor of 36.
- 36 ([tex]\(6^2\)[/tex])
3. List the factors of 36 that are perfect squares: 1, 4, 9, and 36.
4. Identify two factors that multiply together to produce 36:
- [tex]\(4 \times 9 = 36\)[/tex]
- [tex]\(9 \times 4 = 36\)[/tex] (order does not matter as multiplication is commutative)
5. Verify that both 4 and 9 are perfect squares:
- 4 is [tex]\(2^2\)[/tex]
- 9 is [tex]\(3^2\)[/tex]
Taking all these steps into consideration, the two factors of 36 that are perfect squares are [tex]\(4\)[/tex] and [tex]\(9\)[/tex].
