To solve the expression [tex]\((8 \cdot 320)^{\frac{1}{3}}\)[/tex], let's break down the process step-by-step:
1. First, calculate the product inside the parentheses:
[tex]\[
8 \times 320 = 2560
\][/tex]
2. Next, find the cube root of the result:
[tex]\[
(2560)^{\frac{1}{3}}
\][/tex]
3. Compute the cube root of 2560. This calculation gives us approximately:
[tex]\[
(2560)^{\frac{1}{3}} \approx 13.6798
\][/tex]
Now, let's compare this result with each of the given options:
- Option A: [tex]\(8 \sqrt[3]{5}\)[/tex]
[tex]\[
8 \times (5^{\frac{1}{3}}) \approx 8 \times 1.710 \approx 13.6798
\][/tex]
- Option B: 40
[tex]\[
40 \neq 13.6798
\][/tex]
- Option C: 30
[tex]\[
30 \neq 13.6798
\][/tex]
- Option D: [tex]\(10 \sqrt[3]{5}\)[/tex]
[tex]\[
10 \times (5^{\frac{1}{3}}) \approx 10 \times 1.710 \approx 17.0998
\][/tex]
From the calculations, we can see that the expression [tex]\((8 \cdot 320)^{\frac{1}{3}}\)[/tex] is approximately equal to Option A: [tex]\(8 \sqrt[3]{5}\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{8 \sqrt[3]{5}}
\][/tex]
