To simplify ∛54, we start by factoring 54 into its prime factors. The prime factorization of 54 is:
54 = 2 × 3 × 3 × 3 = 2 × 3^3.
When taking the cube root of a number, we can separate perfect cubes from the rest. The cube root of 3^3 is 3, while the cube root of 2 remains under the cube root sign.
Therefore:
∛54 = ∛(2 × 3^3) = ∛(3^3) × ∛(2) = 3∛2.
In this case, the coefficient of 3 outside the cube root represents the cube root of the perfect cube (3^3) extracted from 54.
Looking at the answer choices:
A. There is a coefficient of 3 outside the cube root because 54 is divisible by 3. (This is not a correct reason.) B. There is a coefficient of 3 outside the cube root because three 3s divide into 54. (This is misleading; while it's true there are three 3s, the focus is on the perfect cube, not division.) C. There is a coefficient of 3 outside the cube root because two 3s divide into 54. (This is incorrect; it's the three 3s that matter.) D. There is a coefficient of 3 outside the cube root because 9 divides into 54. (This is true, but it doesn't directly explain why the coefficient of 3 is present.)
Thus, none of the provided options accurately describe the reason for the coefficient of 3 outside the cube root in a straightforward manner based solely on the perfect cube extracted. The best understanding is tied to the existence of the cube of 3 in the factorization of 54. However, the choice indicating 9 could derive from counting factors can be misleading if not understood within the context of cube roots.
Therefore, the best course is to clarify that the coefficient of 3 arises from the perfect cube 3^3 in the prime factorization of 54, but among the given options, D regarding the divisibility of 9 can be interpreted as closely related, though it isn't a perfect explanation.
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