To determine between which two consecutive positive integers the value of [tex]\(\sqrt{40}\)[/tex] lies, let's go through a step-by-step analysis:
1. Finding the Approximate Value of [tex]\(\sqrt{40}\)[/tex]:
- First, we understand that the square root of 40 ([tex]\(\sqrt{40}\)[/tex]) is not a perfect square, so it will be an irrational number somewhere between two integers.
2. Identifying the Consecutive Integers:
- We need to find two consecutive integers such that one is the largest integer smaller than [tex]\(\sqrt{40}\)[/tex] and the other is the smallest integer larger than [tex]\(\sqrt{40}\)[/tex].
3. Evaluating Potential Integers:
- Evaluating the square roots of perfect squares close to 40:
- [tex]\(\sqrt{36} = 6\)[/tex]
- [tex]\(\sqrt{49} = 7\)[/tex]
- From this, we can see that [tex]\(\sqrt{40}\)[/tex] must lie between 6 (since [tex]\(6^2 = 36\)[/tex]) and 7 (since [tex]\(7^2 = 49\)[/tex]).
4. Conclusion:
- Therefore, the value of [tex]\(\sqrt{40}\)[/tex] lies between the integers 6 and 7.
The two consecutive integers between which [tex]\(\sqrt{40}\)[/tex] lies are [tex]\(\boxed{6 \text{ and } 7}\)[/tex].
