To express \(\sqrt{200}\) in its simplest radical form, we need to simplify the expression under the square root. Here are the steps to follow:
1. Factor the number 200: We search for factors of 200 that are perfect squares. The factors are:
[tex]\[
200 = 2 \times 100
\][/tex]
We notice that 100 is a perfect square (since \(10^2 = 100\)).
2. Separate the square root of the product:
We can write the expression as:
[tex]\[
\sqrt{200} = \sqrt{2 \times 100}
\][/tex]
3. Apply the property of square roots: The square root of a product is the product of the square roots. Thus,
[tex]\[
\sqrt{200} = \sqrt{2} \times \sqrt{100}
\][/tex]
4. Simplify the perfect square: Since \(\sqrt{100} = 10\),
[tex]\[
\sqrt{200} = \sqrt{2} \times 10
\][/tex]
5. Combine the terms: This gives us:
[tex]\[
\sqrt{200} = 10\sqrt{2}
\][/tex]
Thus, the simplest radical form of \(\sqrt{200}\) is:
[tex]\[
\boxed{10\sqrt{2}}
\][/tex]
Additionally, the numerical approximation of \(\sqrt{200}\) is approximately:
[tex]\[
10\sqrt{2} \approx 14.142135623730951
\][/tex]
