Which factors of the base would help simplify the expression 1614?(1 point)

Question

Responses

16=4⋅2⋅2
16 equals 4 times 2 times 2

16=8⋅8
16 equals 8 times 8

16=4⋅4
16 equals 4 times 4

16=2⋅2⋅2⋅2
16 equals 2 times 2 times 2 times 2

Answer 1

To simplify the expression \( 16^{14} \), we need to consider the factors of the base, which is 16. Let's analyze the given options with regard to how they can be used to represent or simplify \( 16^{14} \).

1. \( 16 = 4 \cdot 2 \cdot 2 \): This is not a simplification of the base, as 16 equals \( 4 \cdot 4 \) or \( 2^4 \). Use of this representation would complicate calculations.

2. \( 16 = 8 \cdot 8 \): While mathematically correct, this form is not useful for simplification.

3. \( 16 = 4 \cdot 4 \): This representation simplifies the exponentiation a little, as we can express \( 16^{14} \) as \( (4^2)^{14} = 4^{28} \).

4. \( 16 = 2 \cdot 2 \cdot 2 \cdot 2 \): This form represents 16 as \( 2^4 \). Using this, we can rewrite \( 16^{14} \) as \( (2^4)^{14} = 2^{56} \), which is a straightforward simplification.

Out of these options, the most beneficial for simplifying \( 16^{14} \) is \( 16 = 2 \cdot 2 \cdot 2 \cdot 2 \), as it allows us to express the base in terms of its prime factorization and further manipulate the exponent easily.

Therefore, the best choice here is:

\( 16 = 2 \cdot 2 \cdot 2 \cdot 2 \) (i.e., \( 16 = 2^4 \)).