To solve the expression [tex]\((160 \cdot 243)^{\frac{1}{5}}\)[/tex], we need to find the fifth root of the product of 160 and 243.
First, let's combine the numbers inside the parentheses:
[tex]\[
160 \cdot 243
\][/tex]
Next, we take the fifth root of the result:
[tex]\[
(160 \cdot 243)^{\frac{1}{5}}
\][/tex]
Now, we arrive at the numerical value of this expression:
[tex]\[
8.27837796876729
\][/tex]
Given the options:
A. [tex]\(5 \sqrt[5]{5}\)[/tex]
B. 96
C. [tex]\(6 \sqrt[5]{5}\)[/tex]
D. 80
None of the exact forms of the root provided in options A, B, and D match our numerical value. Let's see if option C is indeed the closest equivalent.
Checking option C –
[tex]\[
6 \cdot \sqrt[5]{5}
\][/tex]
Using the numerical value of our previously calculated root, [tex]\(\sqrt[5]{5}\)[/tex] would not accurately give the exact number. Thus, given our precise calculation, none of these values perfectly match.
Therefore, the correct choice based on the closest numerical value is:
[tex]\[
6 \sqrt[5]{5} \approx 8.278
\][/tex]
So the correct answer is:
[tex]\[
C. 6 \sqrt[5]{5}
\][/tex]
