To determine which factors of the base would help simplify the expression [tex]\(16^{\frac{1}{4}}\)[/tex], we need to break down the number 16 into its prime factors.
1. Begin with the given base number, which is 16.
2. Examine how 16 can be factored into prime numbers:
- Notice that [tex]\(16 = 2 \times 2 \times 2 \times 2\)[/tex].
These factors reveal that 16 is composed entirely of the prime number 2, repeated four times. This can also be written as [tex]\(16 = 2^4\)[/tex].
To simplify [tex]\(16^{\frac{1}{4}}\)[/tex]:
[tex]\[
16^{\frac{1}{4}} = (2^4)^{\frac{1}{4}} = 2^{4 \cdot \frac{1}{4}} = 2^1 = 2
\][/tex]
Using the identified prime factors, we were able to simplify the expression.
Out of the given options:
- [tex]\( 16 = 4 \cdot 4 \)[/tex] simplifies as [tex]\(4^2\)[/tex] which is not the prime factor.
- [tex]\( 16 = 4 \cdot 2 \cdot 2 \)[/tex] simplifies using mixed factors but not in prime factor form.
- [tex]\( 16 = 8 \cdot 8 \)[/tex] is incorrect since [tex]\( 8 \cdot 8 = 64 \)[/tex].
- [tex]\( 16 = 2 \cdot 2 \cdot 2 \cdot 2 \)[/tex] is identified as the correct set of prime factors.
Thus, the factors of the base 16 that help simplify [tex]\(16^{\frac{1}{4}}\)[/tex] are:
[tex]\[ 16 = 2 \cdot 2 \cdot 2 \cdot 2 \][/tex]
Therefore, the correct answer is:
[tex]\[ 16 = 2 \cdot 2 \cdot 2 \cdot 2 \][/tex]
