Question

using the commutative property of multiplication, fill in the blanks so that the two algebraic expressions are equivalent:
(14)( )(x) = (7)( )(14)

Answer 1

To make the two algebraic expressions equivalent using the commutative property of multiplication, we can rearrange the factors.

The given expressions are:
\[
(14)( \_ )(x) = (7)( \_ )(14)
\]

We know that the commutative property states that \(a \times b = b \times a\), which means we can rearrange the factors in any way.

Let's fill in the blanks. For the left side, we can choose \(1\) to fill in the first blank, because:
\[
(14)(1)(x) = 14x
\]

On the right side, we can choose \(2\) to fill in the first blank:
\[
(7)(2)(14) = 7 \times 2 \times 14 = 14 \times 7 \times 2 = 14x
\]

However, that doesn't give equivalent factors in \(x\) terms. To make this work, we notice if we take:
\[
(14)(x) = (7)(2)(14)
\]
Then we can set the first blank in the left expression as \(x\), then on the right side we have:
\[
(7)(2)(14) \rightarrow (7)(2) = 14
\]

So we can fill in:
**Left blank:** \(1\) (to accommodate \(x\)),
**Right blank:** \(2\) to keep \(7 \times 2 = 14\).

Final expressions would be:
\[
(14)(1)(x) = (7)(2)(14)
\]

So, filling in the blanks accurately, we have:
\[
(14)(x)(1) = (7)(2)(14)
\]

To simplify this, we could also directly fill:
Thus, both blanks can be filled with \(2\):
\[
(14)(2)(x)=(7)(2)(14)
\]

So the completed answer is \(1\) and \(2\):
\[(14)(1)(x) = (7)(2)(14)\]