Sure, let's work through the question step-by-step.
We need to simplify the expression [tex]\((160 \cdot 243)^{\frac{2}{6}}\)[/tex].
1. Calculate the product of the numbers inside the parenthesis:
[tex]\[
160 \cdot 243
\][/tex]
Which equals:
[tex]\[
160 \cdot 243 = 38880
\][/tex]
2. Simplify the exponent:
The exponent [tex]\(\frac{2}{6}\)[/tex] can be simplified to [tex]\(\frac{1}{3}\)[/tex].
So the expression becomes:
[tex]\[
38880^{\frac{1}{3}}
\][/tex]
3. Calculate the cube root:
The cube root of 38880 is approximately:
[tex]\[
38880^{\frac{1}{3}} \approx 33.87729703971702
\][/tex]
Now we need to match this number, [tex]\(33.87729703971702\)[/tex], with one of the given options:
A. [tex]\(5 \sqrt[3]{5}\)[/tex]
B. 96
C. 80
D. [tex]\(6 \sqrt[3]{5}\)[/tex]
Let's determine the numerical value for each option.
A. [tex]\(5 \sqrt[3]{5}\)[/tex]:
[tex]\[
5 \cdot \sqrt[3]{5} \approx 5 \cdot 1.710 \approx 8.55 \quad (\text{not close})
\][/tex]
B. 96:
[tex]\[
(\text{exact number, no calculation needed, not close})
\][/tex]
C. 80:
[tex]\[
(\text{exact number, no calculation needed, not close})
\][/tex]
D. [tex]\(6 \sqrt[3]{5}\)[/tex]:
[tex]\[
6 \cdot \sqrt[3]{5} \approx 6 \cdot 1.710 \approx 10.26 \quad (\text{not close})
\][/tex]
None of the options given exactly match [tex]\(33.87729703971702\)[/tex], however, choice D ([tex]\(6 \sqrt[3]{5}\)[/tex]) is the closest approximation among them, but it still does not match the calculated value. Probably there is a problem in the options given.
