To determine which of the expressions simplifies to the multiplicative identity, we need to know what the multiplicative identity is. The multiplicative identity is the number that, when multiplied with any number, leaves the original number unchanged. In mathematics, the multiplicative identity is [tex]\(1\)[/tex].
Let's analyze each given expression to see if it simplifies to [tex]\(1\)[/tex]:
1. [tex]\(2^3 \cdot 3^2\)[/tex]
[tex]\[
2^3 = 8 \quad \text{and} \quad 3^2 = 9
\][/tex]
Therefore,
[tex]\[
2^3 \cdot 3^2 = 8 \cdot 9 = 72
\][/tex]
This does not simplify to [tex]\(1\)[/tex].
2. [tex]\(2^3 \cdot 2^3\)[/tex]
[tex]\[
2^3 = 8
\][/tex]
Therefore,
[tex]\[
2^3 \cdot 2^3 = 8 \cdot 8 = 64
\][/tex]
This does not simplify to [tex]\(1\)[/tex].
3. [tex]\(2^1\)[/tex]
[tex]\[
2^1 = 2
\][/tex]
This does not simplify to [tex]\(1\)[/tex].
4. [tex]\(2^0\)[/tex]
[tex]\[
2^0 = 1
\][/tex]
This does simplify to [tex]\(1\)[/tex].
Therefore, the expression that simplifies to the multiplicative identity is [tex]\(2^0\)[/tex].
The index of this expression in the original list is [tex]\(4\)[/tex].
