Let's simplify each exponentiation expression step-by-step and match them to the correct numerical answer:
1. First Expression:
[tex]\[
\frac{2^2 \cdot 3^5}{(2 \cdot \pi)^5}
\][/tex]
Simplifying the numerator:
[tex]\[
2^2 = 4 \quad \text{and} \quad 3^5 = 243 \quad \Rightarrow \quad 4 \cdot 243
\][/tex]
Simplifying the denominator:
[tex]\[
(2 \cdot \pi)^5 = (2 \cdot 3.14)^5 = 248.14^5
\][/tex]
Thus the expression simplifies to:
[tex]\[
\frac{4 \cdot 243}{248.14^5}
\][/tex]
The result is approximately:
[tex]\[
1.0331951895287833 \times 10^{-9}
\][/tex]
2. Second Expression:
[tex]\[
\frac{3^2 \cdot 4^3 \cdot 2^{-1}}{(3 \cdot 4)^2}
\][/tex]
Simplifying the numerator:
[tex]\[
3^2 = 9 \quad \text{and} \quad 4^3 = 64 \quad \text{and} \quad 2^{-1} = 0.5 \quad \Rightarrow \quad 9 \cdot 64 \cdot 0.5
\][/tex]
Simplifying the denominator:
[tex]\[
(3 \cdot 4)^2 = 12^2 = 144
\][/tex]
Thus the expression simplifies to:
[tex]\[
\frac{9 \cdot 64 \cdot 0.5}{144}
\][/tex]
The result is:
[tex]\[
2.0
\][/tex]
3. Third Expression:
[tex]\[
\frac{(3 \cdot 2)^4 \cdot 3^{-3}}{2^3 \cdot 3}
\][/tex]
Simplifying the numerator:
[tex]\[
(3 \cdot 2)^4 = 6^4 \quad \text{and} \quad 3^{-3} = \frac{1}{27} \quad \Rightarrow \quad 6^4 \cdot \frac{1}{27}
\][/tex]
Simplifying the denominator:
[tex]\[
2^3 = 8 \quad \text{and} \quad 3 = 3 \quad \Rightarrow \quad 8 \cdot 3 = 24
\][/tex]
Thus the expression simplifies to:
[tex]\[
\frac{6^4 \cdot \frac{1}{27}}{24}
\][/tex]
The result is:
[tex]\[
2.0
\][/tex]
4. Fourth Expression:
[tex]\[
\frac{3^{-3} \cdot 2^{-3} \cdot 6^3}{(4^0)^2}
\][/tex]
Simplifying the numerator:
[tex]\[
3^{-3} = \frac{1}{27}, \quad 2^{-3} = \frac{1}{8}, \quad \text{and} \quad 6^3 = 216 \quad \Rightarrow \quad \frac{1}{27} \cdot \frac{1}{8} \cdot 216
\][/tex]
Simplifying the denominator:
[tex]\[
(4^0)^2 = 1^2 = 1
\][/tex]
Thus the expression simplifies to:
[tex]\[
\frac{\frac{1}{27} \cdot \frac{1}{8} \cdot 216}{1}
\][/tex]
The result is:
[tex]\[
1.0
\][/tex]
To summarize, the simplified expressions match the following numerical answers:
1. [tex]\(\frac{2^2 \cdot 3^5}{(2 \cdot \pi)^5} \Rightarrow 1.0331951895287833 \times 10^{-9}\)[/tex]
2. [tex]\(\frac{3^2 \cdot 4^3 \cdot 2^{-1}}{(3 \cdot 4)^2} \Rightarrow 2.0\)[/tex]
3. [tex]\(\frac{(3 \cdot 2)^4 \cdot 3^{-3}}{2^3 \cdot 3} \Rightarrow 2.0\)[/tex]
4. [tex]\(\frac{3^{-3} \cdot 2^{-3} \cdot 6^3}{(4^0)^2} \Rightarrow 1.0\)[/tex]
