To find [tex]\(\sqrt{7}\)[/tex], we start by noting that the square root of a number [tex]\(x\)[/tex] is a value [tex]\(y\)[/tex] such that [tex]\(y^2 = x\)[/tex].
### Step-by-Step Solution:
1. Identify the expression: We need to find [tex]\(\sqrt{7}\)[/tex].
2. Estimate the square root:
- Since [tex]\(2^2 = 4\)[/tex] and [tex]\(3^2 = 9\)[/tex], we know that [tex]\(\sqrt{7}\)[/tex] is between 2 and 3.
- It will be closer to 3 because [tex]\(7\)[/tex] is closer to [tex]\(9\)[/tex] than to [tex]\(4\)[/tex].
3. Refine the estimate:
- We can consider the mid-point between 2 and 3, which is 2.5.
- [tex]\(2.5^2 = 6.25\)[/tex], which is less than 7. Hence, [tex]\(\sqrt{7}\)[/tex] is slightly more than 2.5.
- Trying 2.6: [tex]\(2.6^2 = 6.76\)[/tex].
- Trying 2.7: [tex]\(2.7^2 = 7.29\)[/tex].
4. Narrow down the approximation:
- Since [tex]\(6.76 < 7 < 7.29\)[/tex], we narrow it down further between 2.6 and 2.7.
- A refined guess would be around 2.65.
Given these steps and further refining the calculations beyond basic manual methods (which would involve more precise numerical methods for extremely accurate values), we can state:
[tex]\(\sqrt{7} \approx 2.6457513110645907\)[/tex].
This detailed value represents the square root of 7 with high precision.
