Let's classify each of the given real numbers as rational or irrational.
1. [tex]\(5.012121212 \ldots\)[/tex]:
This is a repeating decimal (the sequence "212" repeats indefinitely), which makes it a rational number. Rational numbers include repeating or terminating decimals.
2. [tex]\(\sqrt{1,000}\)[/tex]:
The square root of 1,000 is not a perfect square and does not result in a terminating or repeating decimal. Thus, it is an irrational number.
3. [tex]\(\sqrt[3]{30}\)[/tex]:
The cube root of 30 is not a perfect cube and does not result in a terminating or repeating decimal. Therefore, it is an irrational number.
4. [tex]\(-\frac{5}{250}\)[/tex]:
This can be simplified to [tex]\(-\frac{1}{50}\)[/tex], which is a fraction of integers. Fractions of integers are always rational numbers, so this is a rational number.
5. [tex]\(0.01562138411 \ldots\)[/tex]:
This number does not terminate and does not have a repeating pattern, so it is an irrational number.
6. [tex]\(\sqrt{400}\)[/tex]:
The square root of 400 is 20, which is a whole number. Whole numbers are rational numbers.
Now, we can classify the numbers accordingly:
[tex]\[
\begin{tabular}{|l|l|}
\hline
\text{Rational Numbers} & \text{Irrational Numbers} \\
\hline
5.012121212 \ldots & \sqrt{1,000} \\
-\frac{5}{250} & \sqrt[3]{30} \\
\sqrt{400} & 0.01562138411 \ldots \\
\hline
\end{tabular}
\][/tex]
