Sure, let's solve the expression step-by-step!
The given expression is:
[tex]\[ 32 \cdot \frac{3^{-3}}{2^5} \][/tex]
First, we'll handle each part of the fraction separately.
1. Calculate [tex]\( 3^{-3} \)[/tex]:
The exponent -3 means we take the reciprocal of [tex]\( 3^3 \)[/tex].
[tex]\[ 3^3 = 3 \times 3 \times 3 = 27 \][/tex]
So,
[tex]\[ 3^{-3} = \frac{1}{27} \][/tex]
Therefore,
[tex]\[ 3^{-3} = 0.037037037037037035 \][/tex]
2. Calculate [tex]\( 2^5 \)[/tex]:
This is straightforward exponentiation.
[tex]\[ 2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 \][/tex]
So putting the two results together, we have:
[tex]\[ \frac{3^{-3}}{2^5} = \frac{0.037037037037037035}{32} \][/tex]
3. Simplify the division:
[tex]\[
\frac{0.037037037037037035}{32} = 0.0011574074074074073
\][/tex]
4. Final Multiplication with 32:
Now, multiply the result by 32.
[tex]\[ 32 \cdot 0.0011574074074074073 = 0.037037037037037035 \][/tex]
Hence, the final result is:
[tex]\[ \boxed{0.037037037037037035} \][/tex]
