Answer :
To find [tex]\( F(2) \)[/tex] for the function [tex]\( F(t) = 2 \cdot \frac{1}{2^{3t}} \)[/tex], 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 exponentiation inside the function:
[tex]\[ 3 \cdot 2 = 6 \quad \text{so} \quad 2^{3 \cdot 2} = 2^6 \][/tex]
3. Calculate [tex]\( 2^6 \)[/tex]:
[tex]\[ 2^6 = 64 \][/tex]
4. Plug this result back into the function:
[tex]\[ F(2) = 2 \cdot \frac{1}{64} \][/tex]
5. Simplify the fraction:
[tex]\[ F(2) = \frac{2}{64} \][/tex]
6. Simplify the fraction further:
[tex]\[ \frac{2}{64} = \frac{1}{32} \][/tex]
So [tex]\( F(2) = \frac{1}{32} \)[/tex].
Thus, the correct answer is [tex]\( \boxed{\frac{1}{32}} \)[/tex].
1. Substitute [tex]\( t = 2 \)[/tex] into the function:
[tex]\[ F(2) = 2 \cdot \frac{1}{2^{3 \cdot 2}} \][/tex]
2. Simplify the exponentiation inside the function:
[tex]\[ 3 \cdot 2 = 6 \quad \text{so} \quad 2^{3 \cdot 2} = 2^6 \][/tex]
3. Calculate [tex]\( 2^6 \)[/tex]:
[tex]\[ 2^6 = 64 \][/tex]
4. Plug this result back into the function:
[tex]\[ F(2) = 2 \cdot \frac{1}{64} \][/tex]
5. Simplify the fraction:
[tex]\[ F(2) = \frac{2}{64} \][/tex]
6. Simplify the fraction further:
[tex]\[ \frac{2}{64} = \frac{1}{32} \][/tex]
So [tex]\( F(2) = \frac{1}{32} \)[/tex].
Thus, the correct answer is [tex]\( \boxed{\frac{1}{32}} \)[/tex].
