Question
Which factors of the base would help simplify the expression 1614
16
1
4
?(1 point)
Responses
16=4⋅2⋅2
16
=
4
⋅
2
⋅
2
16 equals 4 times 2 times 2
16=8⋅8
16
=
8
⋅
8
16 equals 8 times 8
16=4⋅4
16
=
4
⋅
4
16 equals 4 times 4
16=2⋅2⋅2⋅2
16
1
4
?(1 point)
Responses
16=4⋅2⋅2
16
=
4
⋅
2
⋅
2
16 equals 4 times 2 times 2
16=8⋅8
16
=
8
⋅
8
16 equals 8 times 8
16=4⋅4
16
=
4
⋅
4
16 equals 4 times 4
16=2⋅2⋅2⋅2
Answers
GPT-4o mini
To simplify the expression \( 16^{\frac{1}{4}} \), it's useful to express the base (16) in terms of its prime factorization. The prime factorization of 16 is \( 2^4 \).
Therefore, we can use any of the equivalent forms of 16 provided in the options to help with simplification.
Here are the options analyzed:
1. **\( 16 = 4 \cdot 2 \cdot 2 \)**: This is valid but not the most direct way to express \( 16 \) for simplification in exponent terms.
2. **\( 16 = 8 \cdot 8 \)**: This is also correct, but it complicates the expression rather than simplifies it in terms of base \( 2 \).
3. **\( 16 = 4 \cdot 4 \)**: This is valid, but it would also convert to \( (2^2) \cdot (2^2) \), and thus doesn't provide a clear simplification.
4. **\( 16 = 2 \cdot 2 \cdot 2 \cdot 2 \)**: While this is technically correct for expressing \( 16 \), it's verbose and does not simplify the expression clearly.
The simplest and most straightforward way to simplify \( 16^{\frac{1}{4}} \) is to express \( 16 \) as \( 2^4 \):
\[
16^{\frac{1}{4}} = (2^4)^{\frac{1}{4}} = 2^{4 \cdot \frac{1}{4}} = 2^1 = 2
\]
Thus, while the options provided all relate to the factorization of \( 16 \), the best choice to simplify the expression effectively would be **\( 16 = 2^4 \)** (if that had been an option). Among the provided options, the one that aligns most closely with a factorization that leads easily into simplifying with exponents would be \( 16 = 4 \cdot 4\) (expressing it as \( (2^2) \cdot (2^2) \)). However, it's worth noting that the ideal method is to work directly with \( 2^4 \).
Therefore, we can use any of the equivalent forms of 16 provided in the options to help with simplification.
Here are the options analyzed:
1. **\( 16 = 4 \cdot 2 \cdot 2 \)**: This is valid but not the most direct way to express \( 16 \) for simplification in exponent terms.
2. **\( 16 = 8 \cdot 8 \)**: This is also correct, but it complicates the expression rather than simplifies it in terms of base \( 2 \).
3. **\( 16 = 4 \cdot 4 \)**: This is valid, but it would also convert to \( (2^2) \cdot (2^2) \), and thus doesn't provide a clear simplification.
4. **\( 16 = 2 \cdot 2 \cdot 2 \cdot 2 \)**: While this is technically correct for expressing \( 16 \), it's verbose and does not simplify the expression clearly.
The simplest and most straightforward way to simplify \( 16^{\frac{1}{4}} \) is to express \( 16 \) as \( 2^4 \):
\[
16^{\frac{1}{4}} = (2^4)^{\frac{1}{4}} = 2^{4 \cdot \frac{1}{4}} = 2^1 = 2
\]
Thus, while the options provided all relate to the factorization of \( 16 \), the best choice to simplify the expression effectively would be **\( 16 = 2^4 \)** (if that had been an option). Among the provided options, the one that aligns most closely with a factorization that leads easily into simplifying with exponents would be \( 16 = 4 \cdot 4\) (expressing it as \( (2^2) \cdot (2^2) \)). However, it's worth noting that the ideal method is to work directly with \( 2^4 \).