To find the square root of [tex]\(57\)[/tex], we want to determine the number which, when multiplied by itself, equals [tex]\(57\)[/tex]. This is denoted by [tex]\(\sqrt{57}\)[/tex].
1. Identify the Problem: We need to find the value of [tex]\(\sqrt{57}\)[/tex]. This is asking us for the non-negative number [tex]\(x\)[/tex] such that [tex]\(x^2 = 57\)[/tex].
2. Conceptual Understanding: The square root operation is essentially the inverse of squaring a number. If [tex]\(x^2 = 57\)[/tex], then [tex]\(x\)[/tex] would be the square root of [tex]\(57\)[/tex].
3. Approximation: Before finding the exact value, it's beneficial to have a rough estimation. We know that [tex]\(7^2 = 49\)[/tex] and [tex]\(8^2 = 64\)[/tex]. Thus, the square root of [tex]\(57\)[/tex] must be between 7 and 8.
4. Precise Calculation:
For an exact value, using more sophisticated methods such as Newton's method or a calculator yields a more precise answer. Calculating [tex]\(\sqrt{57}\)[/tex] precisely, we obtain:
[tex]\[\sqrt{57} \approx 7.54983443527075\][/tex]
5. Verification: To verify, let's square our result:
[tex]\[
(7.54983443527075)^2 = 57.0
\][/tex]
This confirms that our solution is accurate.
Putting it all together, the value of [tex]\(\sqrt{57}\)[/tex] is approximately [tex]\(7.54983443527075\)[/tex].
