To simplify the expression [tex]\( 16^{\frac{1}{4}} \)[/tex], we need to look for a factorization of 16 that will help us understand it better in terms of its base components.
Let's start by breaking down 16:
- First option: [tex]\( 16 = 2 \cdot 2 \cdot 2 \cdot 2 \)[/tex]
This is a complete factorization into prime factors, and it shows that 16 is [tex]\( 2^4 \)[/tex]. When you raise [tex]\( 2^4 \)[/tex] to the power of [tex]\(\frac{1}{4}\)[/tex], you simplify it as follows:
[tex]\[ 16^{\frac{1}{4}} = (2^4)^{\frac{1}{4}} = 2^{4 \cdot \frac{1}{4}} = 2^1 = 2 \][/tex]
So, this factorization is indeed useful and helps simplify the expression.
- Second option: [tex]\( 16 = 4 \cdot 4 \)[/tex]
While this is a valid factorization, it's not as straightforward for simplifying [tex]\( 16^{\frac{1}{4}} \)[/tex] directly, because we are looking into the base-2 factors. Simplification through [tex]\( 4 \cdot 4 \)[/tex] would require recognizing that each 4 is actually [tex]\( 2^2 \)[/tex], making it [tex]\( (2^2) \cdot (2^2) = 2^4 \)[/tex], and then proceeding as above.
- Third option: [tex]\( 16 = 4 \cdot 2 \cdot 2 \)[/tex]
This combination is also valid, but it does not align clearly with the pattern we are seeking related to the base of 2. For our purpose, we need a consistent base that directly simplifies our expression.
- Fourth option: [tex]\( 16 = 8 \cdot 8 \)[/tex]
This is incorrect, as 16 does not equal [tex]\( 8 \cdot 8 \)[/tex] because [tex]\( 8 \cdot 8 = 64 \)[/tex]. Hence, this factorization is not valid and cannot help in simplifying our expression.
Conclusively, the factorization that helps simplify the expression [tex]\( 16^{\frac{1}{4}} \)[/tex] is the first one:
[tex]\[ 16 = 2 \cdot 2 \cdot 2 \cdot 2 \][/tex]
This factorization breaks 16 down into its prime components, making the simplification process straightforward and clear.
