Let's take each item and pair it appropriately based on the mathematical definitions provided.
1. [tex]$w^2 + 5w$[/tex]: This is a combination of terms that can be simplified or factored, and it represents what is known as an expression. An expression is a combination of numbers, variables, and operators that represent a value.
2. the [tex]$w$[/tex] in [tex]$w^2 + 5w$[/tex]: The symbol [tex]$w$[/tex] in the expression stands for a value that can vary, making it a variable. Variables are symbols used to represent unknown values or values that can change.
3. the 2 in [tex]$w^2 + 3x$[/tex]: The number 2 in [tex]$w^2$[/tex] represents how many times the variable [tex]$w$[/tex] is multiplied by itself, i.e., [tex]$w$[/tex] raised to the power of 2. This is known as the exponent. An exponent indicates the power to which a number or variable is raised.
4. the [tex]$\sin w^2 + 5w$[/tex]: This does not form a standard identifiable mathematical term that can be easily simplified or classified in basic algebraic terms, making it Not solvable using Python as it likely refers to a more complex trigonometric expression.
5. the [tex]$w^2$[/tex] or the [tex]$5w$[/tex] in [tex]$w^2 + 5w$[/tex]: Each segment of the expression [tex]$w^2 + 5w$[/tex] representing a product of numbers and variables, separated by a plus or minus sign, is called a term. A term can be a single number, a variable, or a combination of both.
Now, placing these into the provided table:
[tex]\[
\begin{tabular}{|c|c|}
\hline \textbf{Pairs} & \textbf{Classifications} \\
\hline $w^2 + 5w$ & \longrightarrow \text{expression} \\
\hline \text{the $w$ in $w^2 + 5w$} & \longrightarrow \text{variable} \\
\hline \text{the 2 in $w^2 + 3x$} & \longrightarrow \text{exponent} \\
\hline \text{the $\sin w^2 + 5w$} & \longrightarrow \text{Not solvable using Python} \\
\hline \text{the $w^2$ or the $5w$ in $w^2 + 5w$} & \longrightarrow \text{term} \\
\hline
\end{tabular}
\][/tex]
