Let's break down the problem step by step:
1. Compute the cube root of 54:
- The cube root of 54 is approximately 3.7798.
2. Compute the cube root of 16:
- The cube root of 16 is approximately 2.5198.
3. Add these two values:
- Summing these two values gives us approximately 6.2996.
Now we need to determine which of the given options is equivalent to this sum:
(A) [tex]\( 3 \sqrt[3]{6} + 4 \)[/tex]
(B) [tex]\( 3 \sqrt[3]{3} + 2 \sqrt[3]{2} \)[/tex]
(C) 5
(D) [tex]\( 5 \sqrt[3]{2} \)[/tex]
(E) 7
From our calculations, the sum of the cube roots [tex]\( \sqrt[3]{54} + \sqrt[3]{16} \approx 6.2996 \)[/tex].
Comparing this to the options:
- Option (A) [tex]\( 3 \sqrt[3]{6} + 4 \)[/tex] is not close to 6.2996.
- Option (B) [tex]\( 3 \sqrt[3]{3} + 2 \sqrt[3]{2} \)[/tex] is not close to 6.2996.
- Option (C) 5 is less than 6.2996.
- Option (D) [tex]\( 5 \sqrt[3]{2} \)[/tex]:
- The cube root of 2 is approximately [tex]\(1.2599\)[/tex].
- Multiply this by 5 gives approximately [tex]\( 5 \times 1.2599 = 6.2995 \)[/tex], which is very close to 6.2996.
- Option (E) 7 is more than 6.2996.
Given the approximation, option (D) [tex]\( 5 \sqrt[3]{2} \)[/tex] is the value that matches our calculated result most closely. Therefore, the correct answer is:
(D) [tex]\( 5 \sqrt[3]{2} \)[/tex]
