IDNLearn.com connects you with a community of knowledgeable individuals ready to help. Find in-depth and trustworthy answers to all your questions from our experienced community members.
Sagot :
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].
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Find the answers you need at IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.
