To simplify the expression \( 16^{14} \), we need to consider the factors of the base, which is 16. Let's analyze the given options with regard to how they can be used to represent or simplify \( 16^{14} \).
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\( 16 = 4 \cdot 2 \cdot 2 \): This is not a simplification of the base, as 16 equals \( 4 \cdot 4 \) or \( 2^4 \). Use of this representation would complicate calculations.
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\( 16 = 8 \cdot 8 \): While mathematically correct, this form is not useful for simplification.
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\( 16 = 4 \cdot 4 \): This representation simplifies the exponentiation a little, as we can express \( 16^{14} \) as \( (4^2)^{14} = 4^{28} \).
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\( 16 = 2 \cdot 2 \cdot 2 \cdot 2 \): This form represents 16 as \( 2^4 \). Using this, we can rewrite \( 16^{14} \) as \( (2^4)^{14} = 2^{56} \), which is a straightforward simplification.
Out of these options, the most beneficial for simplifying \( 16^{14} \) is \( 16 = 2 \cdot 2 \cdot 2 \cdot 2 \), as it allows us to express the base in terms of its prime factorization and further manipulate the exponent easily.
Therefore, the best choice here is:
\( 16 = 2 \cdot 2 \cdot 2 \cdot 2 \) (i.e., \( 16 = 2^4 \)).
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