Sure, let's evaluate each of the given expressions step-by-step:
1. The first expression is [tex]\(\sqrt{2} \cdot \sqrt[3]{5}\)[/tex]. When we evaluate this, we get the value:
[tex]\[ \sqrt{2} \cdot \sqrt[3]{5} \approx 2.4182711751219577 \][/tex]
2. The second expression is [tex]\(\sqrt[6]{10}\)[/tex]. When we evaluate this, we get the value:
[tex]\[ \sqrt[6]{10} \approx 1.4677992676220695 \][/tex]
3. The third expression is [tex]\(\sqrt[6]{200}\)[/tex]. When we evaluate this, we get the value:
[tex]\[ \sqrt[6]{200} \approx 2.418271175121957 \][/tex]
4. The fourth expression is [tex]\(\sqrt[6]{500}\)[/tex]. When we evaluate this, we get the value:
[tex]\[ \sqrt[6]{500} \approx 2.8172691138478405 \][/tex]
5. The fifth expression is [tex]\(\sqrt[6]{100000}\)[/tex]. When we evaluate this, we get the value:
[tex]\[ \sqrt[6]{100000} \approx 6.812920690579612 \][/tex]
Hence, the complete results for the given expressions are as follows:
[tex]\[
\begin{aligned}
&\sqrt{2} \cdot \sqrt[3]{5} &\approx 2.4182711751219577 \\
&\sqrt[6]{10} &\approx 1.4677992676220695 \\
&\sqrt[6]{200} &\approx 2.418271175121957 \\
&\sqrt[6]{500} &\approx 2.8172691138478405 \\
&\sqrt[6]{100000} &\approx 6.812920690579612 \\
\end{aligned}
\][/tex]
