To determine which expressions are equivalent, let's analyze each one and their mathematical operations:
1. [tex]\((4)(y)\)[/tex]: This denotes the multiplication of 4 and [tex]\(y\)[/tex]. It can be written as:
[tex]\[
(4)(y) = 4 \cdot y
\][/tex]
2. [tex]\(4+y\)[/tex]: This denotes the addition of 4 and [tex]\(y\)[/tex]. Clearly, this expression involves a different operation from multiplication. It can't be equivalent to expressions involving multiplication alone.
3. [tex]\(4 \cdot y\)[/tex]: This explicitly shows the multiplication of 4 and [tex]\(y\)[/tex]. It is already in its final form and quite clear.
4. [tex]\(4y\)[/tex]: In algebra, placing two variables or numbers next to each other typically means multiplication. Thus,
[tex]\[
4y = 4 \cdot y
\][/tex]
Now, let's compare these expressions:
- [tex]\((4)(y)\)[/tex] represents [tex]\(4 \cdot y\)[/tex].
- [tex]\(4 \cdot y\)[/tex] represents itself.
- [tex]\(4y\)[/tex] represents [tex]\(4 \cdot y\)[/tex].
These three expressions [tex]\((4)(y)\)[/tex], [tex]\(4 \cdot y\)[/tex], and [tex]\(4y\)[/tex] all represent the same multiplication operation and thus are equivalent.
However, [tex]\(4 + y\)[/tex] represents addition, which is not the same as multiplying 4 and [tex]\(y\)[/tex]. Therefore, [tex]\(4 + y\)[/tex] is the non-equivalent expression among the given options.
Thus, the answer is the index of the non-equivalent expression, which is:
[tex]\[
1
\][/tex]
