To solve the expression [tex]\((160 \cdot 243)^{\frac{1}{5}}\)[/tex], let's go through the steps in detail.
### Step 1: Multiply 160 and 243
First, we multiply the numbers inside the parentheses:
[tex]\[
160 \cdot 243 = 38880
\][/tex]
### Step 2: Take the 5th root of the product
Next, we need to take the 5th root of the result from step 1. That is:
[tex]\[
38880^{\frac{1}{5}}
\][/tex]
The 5th root of 38880 is approximately:
[tex]\[
8.278
\][/tex]
### Step 3: Compare with the given choices
Now, we need to see which of the given choices is approximately equal to [tex]\(8.278\)[/tex].
1. Choice A: [tex]\(5 \sqrt[5]{5}\)[/tex]
- Let's evaluate [tex]\(5 \sqrt[5]{5}\)[/tex]. The 5th root of 5 is approximately 1.3797.
- Therefore, [tex]\(5 \times 1.3797 \approx 6.8986\)[/tex]
2. Choice B: 96
- This is clearly much larger than our result of [tex]\(8.278\)[/tex].
3. Choice C: [tex]\(6 \sqrt[5]{5}\)[/tex]
- Let's evaluate [tex]\(6 \sqrt[5]{5}\)[/tex]. As before, the 5th root of 5 is approximately 1.3797.
- Therefore, [tex]\(6 \times 1.3797 \approx 8.2784\)[/tex]
4. Choice D: 80
- This is also much larger than our result of [tex]\(8.278\)[/tex].
### Conclusion
After evaluating all the choices, we see that the value given in Choice C ([tex]\(6 \sqrt[5]{5}\)[/tex]) is closest to the value of [tex]\(8.278\)[/tex]. Thus, the correct answer is:
[tex]\[
\boxed{6 \sqrt[5]{5}}
\][/tex]
