To simplify the expression [tex]\( 3^{\frac{\pi}{2}} \cdot 3^{-7} \)[/tex], we need to use the properties of exponents. Specifically, we'll use the property that states [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex].
Let's break down the problem step by step:
1. Combine the exponents:
Given [tex]\( 3^{\frac{\pi}{2}} \cdot 3^{-7} \)[/tex], we add the exponents since the bases are the same:
[tex]\[
3^{\frac{\pi}{2}} \cdot 3^{-7} = 3^{\left(\frac{\pi}{2} + (-7)\right)}
\][/tex]
2. Calculate the combined exponent:
[tex]\[
\frac{\pi}{2} + (-7) = \frac{\pi}{2} - 7
\][/tex]
Numerically, this sum is approximately:
[tex]\[
\frac{\pi}{2} - 7 \approx -5.429203673205103
\][/tex]
3. Simplify the expression using the combined exponent:
[tex]\[
3^{-5.429203673205103}
\][/tex]
4. Evaluate the result:
Using the combined exponent, the base 3 raised to the power of [tex]\( -5.429203673205103 \)[/tex] gives:
[tex]\[
3^{-5.429203673205103} \approx 0.0025680976374452893
\][/tex]
We now compare the simplified result with the given options to see if one matches:
A. [tex]\( \frac{1}{3} \approx 0.3333 \)[/tex]
B. [tex]\( -3 \)[/tex]
C. [tex]\( \frac{1}{243} = 0.00411522633744856 \)[/tex]
D. [tex]\( -\frac{14}{25} \approx -0.56 \)[/tex]
The result [tex]\( 0.0025680976374452893 \)[/tex] is closest to [tex]\( \frac{1}{243} \)[/tex].
Therefore, the correct answer is:
C. [tex]\( \frac{1}{243} \)[/tex]
