To simplify [tex]\(\sqrt{200}\)[/tex], we need to find the prime factorization of 200 and then simplify the square root based on those factors.
1. Prime Factorization:
- The number 200 can be factored as follows:
[tex]\[
200 = 2 \times 100 = 2 \times 2 \times 50 = 2 \times 2 \times 2 \times 25 = 2^3 \times 5^2
\][/tex]
2. Simplifying the Square Root:
- Using the property of square roots, [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex], we can rewrite the square root of 200:
[tex]\[
\sqrt{200} = \sqrt{2^3 \times 5^2}
\][/tex]
- Separate the perfect squares from the factors:
[tex]\[
\sqrt{200} = \sqrt{2^2 \times 2 \times 5^2} = \sqrt{2^2} \times \sqrt{5^2} \times \sqrt{2}
\][/tex]
- Take the square root of the perfect squares:
[tex]\[
\sqrt{200} = 2 \times 5 \times \sqrt{2} = 10 \sqrt{2}
\][/tex]
Thus, [tex]\(\sqrt{200}\)[/tex] in simplest form is [tex]\(10 \sqrt{2}\)[/tex].
Therefore, the correct answer is:
B. [tex]\(10 \sqrt{2}\)[/tex]
