\begin{tabular}{l} Identify the number as rational or irrational. \\ \( 0.345334533345 \ldots \) \\ \hline\end{tabular}

We first recall that a number is rational if and only if its decimal expansion eventually repeats a fixed block of digits (or terminates). In this case, the decimal is given as \[ 0.345334533345\ldots \] A closer look at the digits suggests that the expansion is constructed by concatenating blocks of digits in a pattern. For example, one might observe the following blocks: - The first block is \(345\). - The next block is \(3345\). - The following block could be \(33345\), and so on. Each subsequent block introduces an additional \(3\) at the beginning. This means that no fixed block of digits repeats indefinitely; instead, the length of the non‐repeating section increases without bound. Since the decimal expansion does not settle into a repeating pattern, the number is not rational. Hence, the number is \[ \textbf{irrational}. \]