To find the simplest factorization of 16 that will help simplify the expression [tex]\(16^{\frac{1}{4}}\)[/tex], let's proceed with the factorization that reveals the base raised to a power.
### Step-by-Step Solution:
1. Identify the factors of 16:
- We need a factorization that simplifies the base in such a way that it can be expressed as a single term raised to a power that matches the exponent [tex]\(\frac{1}{4}\)[/tex].
2. List the given factorizations:
- [tex]\(16 = 8 \cdot 8\)[/tex]
- [tex]\(16 = 2 \cdot 2 \cdot 2 \cdot 2\)[/tex]
- [tex]\(16 = 4 \cdot 4\)[/tex]
- [tex]\(16 = 4 \cdot 2 \cdot 2\)[/tex]
3. Analyze the factorizations to find the simplest form:
- To simplify [tex]\(16^{\frac{1}{4}}\)[/tex], we look for a factorization that helps us recognize the base as a power of a smaller number.
4. Evaluate each factorization:
- [tex]\(16 = 8 \cdot 8\)[/tex]
- [tex]\(8 \cdot 8 = 8^2\)[/tex]
- [tex]\(16 = 2 \cdot 2 \cdot 2 \cdot 2\)[/tex]
- Expressed as [tex]\((2^4)\)[/tex]
- [tex]\(16 = 4 \cdot 4\)[/tex]
- [tex]\(4 \cdot 4 = 4^2\)[/tex]
- [tex]\(16 = 4 \cdot 2 \cdot 2\)[/tex]
- This expression doesn’t simplify well directly for [tex]\(16^{\frac{1}{4}}\)[/tex].
5. Select the simplest and most appropriate factorization for [tex]\(16^{\frac{1}{4}}\)[/tex]:
- Among all factorizations, [tex]\(16 = 2 \cdot 2 \cdot 2 \cdot 2\)[/tex] directly reveals [tex]\(2^4\)[/tex]. This is ideal because:
- We can write [tex]\(16\)[/tex] as [tex]\(2^4\)[/tex].
- Therefore, [tex]\(16^{\frac{1}{4}} = (2^4)^{\frac{1}{4}} = 2^{(4 \cdot \frac{1}{4})} = 2^1 = 2\)[/tex].
### Conclusion:
The correct factorization that simplifies [tex]\(16^{\frac{1}{4}}\)[/tex] is:
[tex]\[ 16 = 2 \cdot 2 \cdot 2 \cdot 2 \][/tex]
So, the choice is:
[tex]\[ \boxed{16 = 2 \cdot 2 \cdot 2 \cdot 2} \][/tex]
