To simplify the expression \( 16^{14} \), the choice of factors for the base can significantly affect the simplification process. We want to express \( 16 \) in terms of its prime factors.
The prime factorization of \( 16 \) is:
- \( 16 = 2^4 \)
Using this factorization:
- \( 16^{14} = (2^4)^{14} = 2^{4 \cdot 14} = 2^{56} \)
Now let's examine each of the provided options for how they might help to simplify \( 16 \):
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\( 16 = 4 \cdot 4 \): This is valid, but it doesn't lead directly to a prime factorization that simplifies the expression effectively.
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\( 16 = 2 \cdot 2 \cdot 2 \cdot 2 \): This is also valid and directly shows the prime factorization.
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\( 16 = 4 \cdot 2 \cdot 2 \): This representation might not be as straightforward for simplification compared to the full prime factorization.
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\( 16 = 8 \cdot 8 \): While mathematically correct, this also does not lead to a simplification that uses primes.
Among these options, the best one that helps simplify the expression is:
- \( 16 = 2 \cdot 2 \cdot 2 \cdot 2 \), as it leads to the simplest form using prime factorization.
Thus, the most useful option for simplifying the expression \( 16^{14} \) is: 16 = 2 ⋅ 2 ⋅ 2 ⋅ 2 (16 equals 2 times 2 times 2 times 2).