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To simplify the expression [tex]\( 16^{\frac{1}{4}} \)[/tex], we need to explore the factorization of the base, which is 16. Let's break down 16 into its factors and examine which one would help simplify the expression [tex]\( 16^{\frac{1}{4}} \)[/tex].
1. First Factorization: [tex]\( 16 = 2 \cdot 2 \cdot 2 \cdot 2 \)[/tex]
[tex]\[ 16^{\frac{1}{4}} = (2 \cdot 2 \cdot 2 \cdot 2)^{\frac{1}{4}} \][/tex]
We know that [tex]\((a \cdot b)^{n} = a^{n} \cdot b^{n}\)[/tex]. Thus,
[tex]\[ 16^{\frac{1}{4}} = (2^{4})^{\frac{1}{4}} \][/tex]
[tex]\[ 16^{\frac{1}{4}} = 2^{4 \cdot \frac{1}{4}} \][/tex]
[tex]\[ 16^{\frac{1}{4}} = 2^{1} \][/tex]
[tex]\[ 16^{\frac{1}{4}} = 2 \][/tex]
2. Second Factorization: [tex]\( 16 = 4 \cdot 4 \)[/tex]
[tex]\[ 16^{\frac{1}{4}} = (4 \cdot 4)^{\frac{1}{4}} \][/tex]
Again, we can apply the rule [tex]\((a \cdot b)^{n} = a^{n} \cdot b^{n}\)[/tex]. Thus,
[tex]\[ 16^{\frac{1}{4}} = 4^{\frac{1}{4}} \cdot 4^{\frac{1}{4}} \][/tex]
Since [tex]\( 4 = 2^{2} \)[/tex],
[tex]\[ 4^{\frac{1}{4}} = (2^{2})^{\frac{1}{4}} \][/tex]
[tex]\[ 4^{\frac{1}{4}} = 2^{2 \cdot \frac{1}{4}} \][/tex]
[tex]\[ 4^{\frac{1}{4}} = 2^{\frac{1}{2}} \][/tex]
Therefore,
[tex]\[ 4^{\frac{1}{4}} \cdot 4^{\frac{1}{4}} = 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{\frac{1}{2} + \frac{1}{2}} = 2^{1} \][/tex]
[tex]\[ 16^{\frac{1}{4}} = 2 \][/tex]
3. Third Factorization: [tex]\( 16 = 8 \cdot 8 \)[/tex]
[tex]\[ 16^{\frac{1}{4}} = (8 \cdot 8)^{\frac{1}{4}} \][/tex]
Using the rule [tex]\((a \cdot b)^{n} = a^{n} \cdot b^{n}\)[/tex],
[tex]\[ 16^{\frac{1}{4}} = 8^{\frac{1}{4}} \cdot 8^{\frac{1}{4}} \][/tex]
Since [tex]\( 8 = 2^{3} \)[/tex],
[tex]\[ 8^{\frac{1}{4}} = (2^{3})^{\frac{1}{4}} \][/tex]
[tex]\[ 8^{\frac{1}{4}} = 2^{3 \cdot \frac{1}{4}} \][/tex]
[tex]\[ 8^{\frac{1}{4}} = 2^{\frac{3}{4}} \][/tex]
Therefore,
[tex]\[ 8^{\frac{1}{4}} \cdot 8^{\frac{1}{4}} = 2^{\frac{3}{4}} \cdot 2^{\frac{3}{4}} = 2^{\frac{3}{4} + \frac{3}{4}} = 2^{\frac{6}{4}} = 2^{\frac{3}{2}} \][/tex]
In this case, the factorization using [tex]\( 8 \cdot 8 \)[/tex] does not directly simplify to [tex]\( 2 \)[/tex].
4. Fourth Factorization: [tex]\( 16 = 4 \cdot 2 \cdot 2 \)[/tex]
[tex]\[ 16^{\frac{1}{4}} = (4 \cdot 2 \cdot 2)^{\frac{1}{4}} \][/tex]
Applying the rule [tex]\((a \cdot b)^{n} = a^{n} \cdot b^{n}\)[/tex],
[tex]\[ 16^{\frac{1}{4}} = 4^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \][/tex]
We already know from the second factorization that [tex]\( 4^{\frac{1}{4}} = 2^{\frac{1}{2}} \)[/tex]. Thus,
[tex]\[ 16^{\frac{1}{4}} = 2^{\frac{1}{2}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \][/tex]
[tex]\[ 16^{\frac{1}{4}} = 2^{\frac{1}{2} + \frac{1}{4} + \frac{1}{4}} = 2^{\frac{1}{2} + \frac{2}{4}} = 2^{\frac{1}{2} + \frac{1}{2}} = 2^{1} \][/tex]
[tex]\[ 16^{\frac{1}{4}} = 2 \][/tex]
Thus, the factorization [tex]\( 16 = 2 \cdot 2 \cdot 2 \cdot 2 \)[/tex] simplifies the expression [tex]\( 16^{\frac{1}{4}} \)[/tex] to [tex]\( 2 \)[/tex]. Therefore, the relevant factors for simplifying [tex]\( 16^{\frac{1}{4}} \)[/tex] are [tex]\( 2, 2, 2, 2 \)[/tex].
1. First Factorization: [tex]\( 16 = 2 \cdot 2 \cdot 2 \cdot 2 \)[/tex]
[tex]\[ 16^{\frac{1}{4}} = (2 \cdot 2 \cdot 2 \cdot 2)^{\frac{1}{4}} \][/tex]
We know that [tex]\((a \cdot b)^{n} = a^{n} \cdot b^{n}\)[/tex]. Thus,
[tex]\[ 16^{\frac{1}{4}} = (2^{4})^{\frac{1}{4}} \][/tex]
[tex]\[ 16^{\frac{1}{4}} = 2^{4 \cdot \frac{1}{4}} \][/tex]
[tex]\[ 16^{\frac{1}{4}} = 2^{1} \][/tex]
[tex]\[ 16^{\frac{1}{4}} = 2 \][/tex]
2. Second Factorization: [tex]\( 16 = 4 \cdot 4 \)[/tex]
[tex]\[ 16^{\frac{1}{4}} = (4 \cdot 4)^{\frac{1}{4}} \][/tex]
Again, we can apply the rule [tex]\((a \cdot b)^{n} = a^{n} \cdot b^{n}\)[/tex]. Thus,
[tex]\[ 16^{\frac{1}{4}} = 4^{\frac{1}{4}} \cdot 4^{\frac{1}{4}} \][/tex]
Since [tex]\( 4 = 2^{2} \)[/tex],
[tex]\[ 4^{\frac{1}{4}} = (2^{2})^{\frac{1}{4}} \][/tex]
[tex]\[ 4^{\frac{1}{4}} = 2^{2 \cdot \frac{1}{4}} \][/tex]
[tex]\[ 4^{\frac{1}{4}} = 2^{\frac{1}{2}} \][/tex]
Therefore,
[tex]\[ 4^{\frac{1}{4}} \cdot 4^{\frac{1}{4}} = 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{\frac{1}{2} + \frac{1}{2}} = 2^{1} \][/tex]
[tex]\[ 16^{\frac{1}{4}} = 2 \][/tex]
3. Third Factorization: [tex]\( 16 = 8 \cdot 8 \)[/tex]
[tex]\[ 16^{\frac{1}{4}} = (8 \cdot 8)^{\frac{1}{4}} \][/tex]
Using the rule [tex]\((a \cdot b)^{n} = a^{n} \cdot b^{n}\)[/tex],
[tex]\[ 16^{\frac{1}{4}} = 8^{\frac{1}{4}} \cdot 8^{\frac{1}{4}} \][/tex]
Since [tex]\( 8 = 2^{3} \)[/tex],
[tex]\[ 8^{\frac{1}{4}} = (2^{3})^{\frac{1}{4}} \][/tex]
[tex]\[ 8^{\frac{1}{4}} = 2^{3 \cdot \frac{1}{4}} \][/tex]
[tex]\[ 8^{\frac{1}{4}} = 2^{\frac{3}{4}} \][/tex]
Therefore,
[tex]\[ 8^{\frac{1}{4}} \cdot 8^{\frac{1}{4}} = 2^{\frac{3}{4}} \cdot 2^{\frac{3}{4}} = 2^{\frac{3}{4} + \frac{3}{4}} = 2^{\frac{6}{4}} = 2^{\frac{3}{2}} \][/tex]
In this case, the factorization using [tex]\( 8 \cdot 8 \)[/tex] does not directly simplify to [tex]\( 2 \)[/tex].
4. Fourth Factorization: [tex]\( 16 = 4 \cdot 2 \cdot 2 \)[/tex]
[tex]\[ 16^{\frac{1}{4}} = (4 \cdot 2 \cdot 2)^{\frac{1}{4}} \][/tex]
Applying the rule [tex]\((a \cdot b)^{n} = a^{n} \cdot b^{n}\)[/tex],
[tex]\[ 16^{\frac{1}{4}} = 4^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \][/tex]
We already know from the second factorization that [tex]\( 4^{\frac{1}{4}} = 2^{\frac{1}{2}} \)[/tex]. Thus,
[tex]\[ 16^{\frac{1}{4}} = 2^{\frac{1}{2}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \][/tex]
[tex]\[ 16^{\frac{1}{4}} = 2^{\frac{1}{2} + \frac{1}{4} + \frac{1}{4}} = 2^{\frac{1}{2} + \frac{2}{4}} = 2^{\frac{1}{2} + \frac{1}{2}} = 2^{1} \][/tex]
[tex]\[ 16^{\frac{1}{4}} = 2 \][/tex]
Thus, the factorization [tex]\( 16 = 2 \cdot 2 \cdot 2 \cdot 2 \)[/tex] simplifies the expression [tex]\( 16^{\frac{1}{4}} \)[/tex] to [tex]\( 2 \)[/tex]. Therefore, the relevant factors for simplifying [tex]\( 16^{\frac{1}{4}} \)[/tex] are [tex]\( 2, 2, 2, 2 \)[/tex].
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