Question

Which factors of the base would help simplify the expression 16 1/4
2•2•2•2
4•2•2
4•4
8•8

Answer 1

To simplify the expression \( 16 \frac{1}{4} \) with the base numbers given, we can start by converting \( 16 \frac{1}{4} \) into an improper fraction:

\[
16 \frac{1}{4} = 16 + \frac{1}{4} = \frac{64}{4} + \frac{1}{4} = \frac{65}{4}
\]

Now, let’s analyze the bases provided:

1. \(2 \cdot 2 \cdot 2 \cdot 2 = 2^4 = 16\)
2. \(4 \cdot 2 \cdot 2 = 4 \cdot 4 = 16\)
3. \(4 \cdot 4 = 16\)
4. \(8 \cdot 8 = 64\)

Next, let's express \( \frac{65}{4} \) in terms of the base factorizations:

From the bases given:
- The base of \(4\) can be expressed as \(2^2\).
- The base of \(8\) can be expressed as \(2^3\).

Since we want to find factors that help simplify the fraction, we note that \(4 = 2^2\) and \(16 = 2^4\).

Now we can rewrite \( \frac{65}{4} \) as:

\[
\frac{65}{4} = \frac{65}{2^2}
\]

While \(65\) does not further simplify with respect to \(2\), we can also keep \(4\) as a base of \(2\).

To summarize, the factors that help in simplifying based on the powers of \(2\) are:

- \(2 \cdot 2 \cdot 2 \cdot 2\) (from \(16\))
- \(4 \cdot 2 \cdot 2\)
- \(4 \cdot 4\)

However, \(8 \cdot 8\) would be less useful because it represents an altogether different power (does not relate to simplifying \( \frac{65}{4}\)).

Ultimately, the base of \(4\) (\(2^2\)) and \(16\) (\(2^4\)) can be effectively utilized to express and simplify the component of \( \frac{65}{4} \) if connected with additional factors or other expressions.