Explanation

(d)
$\left\lbrack (0.16)^{\log_{0.25}\left( \frac{1}{3} + \frac{1}{3^{2}} + \frac{1}{3^{3}} + \ldots \right)} \right\rbrack^{\frac{1}{2}}$
$= \left\lbrack (0.16)^{\ \log_{0.25}\left( \frac{\frac{1}{3}}{1 - \frac{1}{3}} \right)} \right\rbrack^{\frac{1}{2}}$
$= \left\lbrack (0.16)^{\log_{(0.5)^{2}}{0.5}} \right\rbrack^{\frac{1}{2}}$
$= \left\lbrack (0.16)^{\frac{1}{2}} \right\rbrack^{\frac{1}{2}} = (0.4)^{\frac{1}{2}}$
$= \frac{2}{\sqrt{10}}$