Let's solve [tex]\(200 + 8 \sqrt{200} - 2 \sqrt{363}\)[/tex] step by step.
1. Calculate [tex]\(\sqrt{200}\)[/tex] and [tex]\(\sqrt{363}\)[/tex]:
[tex]\[
\sqrt{200} \approx 14.1421
\][/tex]
[tex]\[
\sqrt{363} \approx 19.0526
\][/tex]
2. Multiply these square roots by the appropriate coefficients:
[tex]\[
8 \sqrt{200} = 8 \times 14.1421 \approx 113.1371
\][/tex]
[tex]\[
2 \sqrt{363} = 2 \times 19.0526 \approx 38.1051
\][/tex]
3. Substitute these values back into the original expression:
[tex]\[
200 + 8 \sqrt{200} - 2 \sqrt{363} \approx 200 + 113.1371 - 38.1051
\][/tex]
4. Perform the final addition and subtraction:
[tex]\[
200 + 113.1371 - 38.1051 \approx 275.0320
\][/tex]
Thus, the final value of [tex]\(200 + 8 \sqrt{200} - 2 \sqrt{363}\)[/tex] is approximately [tex]\(275.0320\)[/tex].
