Let's address the problem step-by-step to determine which option is equal to the expression [tex]\((8 \cdot 320)^{\frac{1}{3}}\)[/tex].
Step 1: Evaluate the expression [tex]\((8 \cdot 320)^{\frac{1}{3}}\)[/tex].
Starting with the expression inside the parentheses:
[tex]\[ 8 \cdot 320 = 2560 \][/tex]
Step 2: Take the cube root of the result obtained in Step 1.
[tex]\[ (2560)^{\frac{1}{3}} \approx 13.679807573413575 \][/tex]
From this evaluation, we see that:
[tex]\[ (8 \cdot 320)^{\frac{1}{3}} \approx 13.679807573413575 \][/tex]
Step 3: Evaluate the given answer choices one by one and see which one matches the calculated value.
Option A: [tex]\( 10 \sqrt[3]{5} \)[/tex]
Calculate [tex]\( 10 \sqrt[3]{5} \)[/tex]:
[tex]\[ 10 \times (5)^{\frac{1}{3}} \approx 10 \times 1.710 = 17.09975946676697 \][/tex]
This value is approximately 17.1, which does not match 13.679807573413575.
Option B: 40
This value is clearly 40, which does not match 13.679807573413575.
Option C: [tex]\( 8 \sqrt[3]{5} \)[/tex]
Calculate [tex]\( 8 \sqrt[3]{5} \)[/tex]:
[tex]\[ 8 \times (5)^{\frac{1}{3}} \approx 8 \times 1.710 = 13.679807573413575 \][/tex]
This value is approximately 13.679807573413575, which matches our initial calculation exactly.
Option D: 30
This value is clearly 30, which does not match 13.679807573413575.
Based on the detailed evaluations, the correct answer is:
c. [tex]\( 8 \sqrt[3]{5} \)[/tex]
