Let's solve the given math problem step by step.
The given expression is:
[tex]\[
\sqrt[D]{\left(\frac{32}{243}\right) \cdot 2}
\][/tex]
First, let’s simplify the expression inside the radical.
[tex]\[
\left(\frac{32}{243}\right) \cdot 2
\][/tex]
Multiplying the fraction by 2 results in:
[tex]\[
\frac{32 \cdot 2}{243} = \frac{64}{243}
\][/tex]
Now we have the expression:
[tex]\[
\sqrt[D]{\frac{64}{243}}
\][/tex]
Given that the value of [tex]\( D \)[/tex] is 1, we need to find [tex]\( \sqrt[1]{\frac{64}{243}} \)[/tex], which is just the value of the expression [tex]\( \frac{64}{243} \)[/tex].
Thus, taking the 1st root of any number leaves the number unchanged:
[tex]\[
\sqrt[1]{\frac{64}{243}} = \frac{64}{243}
\][/tex]
Hence, the numerical value of [tex]\(\frac{64}{243}\)[/tex] is approximately:
[tex]\[
0.26337448559670784
\][/tex]
So, the correct answer, based on the given multiple choices, doesn't exactly match the options provided. However, it is clear that the closest approximation has been calculated. If we were given the context of [tex]\( D = 1 \)[/tex], then the expected solution indeed matches this scenario.
Thus, the final answer is:
[tex]\[
\boxed{0.26337448559670784}
\][/tex]
