To find [tex]\( F(2) \)[/tex] using the function [tex]\( F(t) = 2 \cdot \frac{1}{2^{3t}} \)[/tex], we follow these steps:
1. Substitute [tex]\( t = 2 \)[/tex] into the function:
[tex]\[
F(2) = 2 \cdot \frac{1}{2^{3 \cdot 2}}
\][/tex]
2. Simplify the exponent:
The exponent in the denominator [tex]\( 3 \cdot 2 \)[/tex] simplifies to:
[tex]\[
3 \cdot 2 = 6
\][/tex]
This means the expression now looks like:
[tex]\[
F(2) = 2 \cdot \frac{1}{2^6}
\][/tex]
3. Evaluate the denominator:
Calculate [tex]\( 2^6 \)[/tex]:
[tex]\[
2^6 = 64
\][/tex]
4. Put it all together:
Now the expression becomes:
[tex]\[
F(2) = 2 \cdot \frac{1}{64} = 2 \cdot \frac{1}{64} = \frac{2}{64}
\][/tex]
5. Simplify [tex]\( \frac{2}{64} \)[/tex]:
Simplify [tex]\( \frac{2}{64} \)[/tex] by dividing the numerator and denominator by 2:
[tex]\[
\frac{2}{64} = \frac{1}{32}
\][/tex]
Thus, we arrive at the final value of [tex]\( F(2) \)[/tex]:
[tex]\[
\boxed{\frac{1}{32}}
\][/tex]
Therefore, the correct answer is:
D. [tex]\( \frac{1}{32} \)[/tex]
