Which is the *smallest* of the following numbers?

\(\left(\sqrt{3}\right)^3\),

\(\log_3 \left(9^2\right)\),

\(\left( 3\sin{\dfrac{\pi}{3}}\right) ^2\),

\(\log_2 \left(\log_2 \left(8^5\right)\right)\).

First we note that the value for b. is \(\log_3 (9^2) = 2\log_3 9 = 2\times2 = 4\).

In comparison with this,

a: \(\quad(\sqrt{3})^3 = 3 \sqrt{3} > 4\), since \(\sqrt{3} > 1.5\).

c: \(\quad\left(3\sin{\dfrac{\pi}{3}}\right)^2 = \left(\dfrac{3 \sqrt{3}}{2} \right)^2 = \frac{27}{4} > 4\).

d: \(\quad\log_2 (\log_2 (8^5)) = \log_2 (5\log_2 8) = \log_2 15 < \log_2 16 = 4\).

Hence the answer is d.