To simplify the expression \( 16^{\frac{1}{4}} \), it's useful to express the base (16) in terms of its prime factorization. The prime factorization of 16 is \( 2^4 \).
Therefore, we can use any of the equivalent forms of 16 provided in the options to help with simplification.
Here are the options analyzed:
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\( 16 = 4 \cdot 2 \cdot 2 \): This is valid but not the most direct way to express \( 16 \) for simplification in exponent terms.
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\( 16 = 8 \cdot 8 \): This is also correct, but it complicates the expression rather than simplifies it in terms of base \( 2 \).
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\( 16 = 4 \cdot 4 \): This is valid, but it would also convert to \( (2^2) \cdot (2^2) \), and thus doesn't provide a clear simplification.
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\( 16 = 2 \cdot 2 \cdot 2 \cdot 2 \): While this is technically correct for expressing \( 16 \), it's verbose and does not simplify the expression clearly.
The simplest and most straightforward way to simplify \( 16^{\frac{1}{4}} \) is to express \( 16 \) as \( 2^4 \):
\[ 16^{\frac{1}{4}} = (2^4)^{\frac{1}{4}} = 2^{4 \cdot \frac{1}{4}} = 2^1 = 2 \]
Thus, while the options provided all relate to the factorization of \( 16 \), the best choice to simplify the expression effectively would be \( 16 = 2^4 \) (if that had been an option). Among the provided options, the one that aligns most closely with a factorization that leads easily into simplifying with exponents would be \( 16 = 4 \cdot 4\) (expressing it as \( (2^2) \cdot (2^2) \)). However, it's worth noting that the ideal method is to work directly with \( 2^4 \).
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