Which is the smallest of these values?

\(\log_{10}\pi\),

\(\sqrt{\log_{10}(\pi^2)}\),

\(\left(\dfrac{1}{\log_{10}\pi}\right)^3\),

\(\dfrac{1}{\log_{10}\sqrt{\pi}}\).

Let \(L = \log_{10}\pi\). Since \(\pi < 10\), we have \(L < 1\). Then

\(\log_{10}\pi = L\);

\(\sqrt{\log_{10}(\pi^2)}= \sqrt{2\log_{10}\pi} = \sqrt{2L} > \sqrt{L \times L} = L\);

\(\left(\dfrac{1}{\log_{10}\pi}\right)^3 = L^{-3} > 1\);

\(\dfrac{1}{\log_{10}\sqrt{\pi}} = \dfrac{1}{\dfrac{1}{2}\log_{10}\pi} = \dfrac{2}{L} > 2\).

So the smallest value is \(L\), and the answer is (a).