Cho các số thực dương $a$, $b$ thỏa mãn $\sqrt{\log a}+\sqrt{\log...

Ta có: $\sqrt{\log a}+\sqrt{\log b}+\log \sqrt{a}+\log \sqrt{b}=100$.
$\Leftrightarrow \log a+\log b+2\sqrt{\log a}+2\sqrt{\log b}=200$ $\Leftrightarrow {{\left( \sqrt{\log a}+1 \right)}^{2}}+{{\left( \sqrt{\log b}+1 \right)}^{2}}=202=81+121$ $\left( * \right)$
Mà $\sqrt{\log a}$, $\sqrt{\log b}$, $\log \sqrt{a}$, $\log \sqrt{b}$ đều là các số nguyên dương nên
$\left( * \right)\Leftrightarrow \left[ \begin{matrix}
\left\{ \begin{matrix}
\sqrt{\log a}+1=9 \\
\sqrt{\log b}+1=11 \\
\end{matrix} \right. \\
\left\{ \begin{matrix}
\sqrt{\log a}+1=11 \\
\sqrt{\log b}+1=9 \\
\end{matrix} \right. \\
\end{matrix} \right.\Leftrightarrow \left[ \begin{matrix}
\left\{ \begin{matrix}
\log a=64 \\
\log b=100 \\
\end{matrix} \right. \\
\left\{ \begin{matrix}
\log a=100 \\
\log b=64 \\
\end{matrix} \right. \\
\end{matrix} \right.\Leftrightarrow \left[ \begin{matrix}
\left\{ \begin{matrix}
a={{10}^{64}} \\
b={{10}^{100}} \\
\end{matrix} \right. \\
\left\{ \begin{matrix}
a={{10}^{100}} \\
b={{10}^{64}} \\
\end{matrix} \right. \\
\end{matrix} \right.$
Vậy: $P=ab={{10}^{64}}{{.10}^{100}}={{10}^{164}}.$