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Rational & Irrational Numbers

Read each square below. If the number in the box is a rational number, color it green. If the number in the box is an irrational number, color it yellow.

\begin{tabular}{|c|c|c|c|c|}
\hline
5000 & \begin{tabular}{c}
perfect \\
square \\
[tex]$\sqrt{9}=3$[/tex]
\end{tabular} & \begin{tabular}{c}
terminates \\
-4 \\
[tex]$R$[/tex]
\end{tabular} & \begin{tabular}{c}
repeats \\
[tex]$3.\overline{2}$[/tex] \\
[tex]$R$[/tex]
\end{tabular} & \begin{tabular}{c}
3.421553...
\end{tabular} \\
\hline
\begin{tabular}{c}
[tex]$\sqrt{32}$[/tex]
\end{tabular} & \begin{tabular}{c}
perfect \\
square \\
[tex]$\sqrt{81}$[/tex]
\end{tabular} & \begin{tabular}{c}
terminates \\
0 \\
[tex]$R$[/tex]
\end{tabular} & \begin{tabular}{c}
not terminates \\
[tex]$\pi$[/tex]
\end{tabular} & \begin{tabular}{c}
12.2 \\
[tex]$R$[/tex]
\end{tabular} \\
\hline
\begin{tabular}{c}
[tex]$\frac{1}{8}$[/tex]
\end{tabular} & -22.3 & [tex]$\sqrt{144}$[/tex] & [tex]$2^5$[/tex] & [tex]$\sqrt{21}$[/tex] \\
\hline
4.5 & \begin{tabular}{c}
1 \\
[tex]$R$[/tex]
\end{tabular} & [tex]$\sqrt{36}$[/tex] & \begin{tabular}{c}
Terminates \\
2
\end{tabular} & \begin{tabular}{c}
5.320...
\end{tabular} \\
\hline
\begin{tabular}{c}
[tex]$\frac{\sqrt{100}}{2}$[/tex]
\end{tabular} & [tex]$\sqrt{40}$[/tex] & \begin{tabular}{c}
[tex]$5 \frac{2}{5}$[/tex]
\end{tabular} & [tex]$0.\overline{132}$[/tex] & \begin{tabular}{c}
\end{tabular} \\
\hline
\end{tabular}

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Answer :

To determine whether each number is rational or irrational in the given table, follow the definitions of rational and irrational numbers:

Rational Numbers: Numbers that can be expressed as a fraction [tex]\(\frac{a}{b}\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers, and [tex]\(b \neq 0\)[/tex]. This includes integers, terminating decimals, and repeating decimals.

Irrational Numbers: Numbers that cannot be expressed as a simple fraction. These numbers have non-terminating, non-repeating decimal expansions.

Let's go through each number step-by-step:

1. 5000: Rational - It is an integer.

2. [tex]\(\sqrt{9} = 3\)[/tex]: Rational - [tex]\(3\)[/tex] is an integer.

3. -4: Rational - It is an integer.

4. 3.2 (Repeats): Rational - A repeating decimal can be expressed as a fraction.

5. 3.421553...: Irrational - A decimal that does not terminate or repeat.

6. [tex]\(\sqrt{32}\)[/tex]: Irrational - Cannot be simplified to an integer.

7. [tex]\(\sqrt{81} = 9\)[/tex]: Rational - [tex]\(9\)[/tex] is an integer.

8. 0: Rational - It is an integer.

9. [tex]\(\pi\)[/tex]: Irrational - [tex]\(\pi\)[/tex] has a non-terminating, non-repeating decimal expansion.

10. 12.2: Rational - A terminating decimal.

11. [tex]\(\frac{1}{8}\)[/tex]: Rational - It is a fraction.

12. -22.3: Rational - A terminating decimal.

13. [tex]\(\sqrt{144} = 12\)[/tex]: Rational - [tex]\(12\)[/tex] is an integer.

14. [tex]\(2^5 = 32\)[/tex]: Rational - [tex]\(32\)[/tex] is an integer.

15. [tex]\(\sqrt{21}\)[/tex]: Irrational - Cannot be simplified to an integer.

16. 4.5: Rational - A terminating decimal.

17. 1: Rational - It is an integer.

18. [tex]\(\sqrt{36} = 6\)[/tex]: Rational - [tex]\(6\)[/tex] is an integer.

19. 2 (Terminates): Rational - A terminating number.

20. 5.320..: Irrational - A non-terminating, non-repeating decimal.

21. [tex]\(\frac{\sqrt{100}}{2} = 5\)[/tex]: Rational - [tex]\(5\)[/tex] is an integer.

22. [tex]\(\sqrt{40}\)[/tex]: Irrational - Cannot be simplified to an integer.

23. 5 [tex]\(\frac{2}{5}\)[/tex]: Rational - It can be expressed as an improper fraction.

24. 0.132 (Repeats): Rational - A repeating decimal can be expressed as a fraction.

According to the criteria:

- Rational Numbers:
1. 5000
2. [tex]\(\sqrt{9} = 3\)[/tex]
3. -4
4. 3.2 (Repeats)
5. [tex]\(\sqrt{81} = 9\)[/tex]
6. 0
7. 12.2
8. [tex]\(\frac{1}{8}\)[/tex]
9. -22.3
10. [tex]\(\sqrt{144} = 12\)[/tex]
11. [tex]\(2^5 = 32\)[/tex]
12. 4.5
13. 1
14. [tex]\(\sqrt{36} = 6\)[/tex]
15. 2 (Terminates)
16. [tex]\(\frac{\sqrt{100}}{2} = 5\)[/tex]
17. 5 [tex]\(\frac{2}{5}\)[/tex]
18. 0.132 (Repeats)

- Irrational Numbers:
1. 3.421553...
2. [tex]\(\sqrt{32}\)[/tex]
3. [tex]\(\pi\)[/tex]
4. [tex]\(\sqrt{21}\)[/tex]
5. 5.320..
6. [tex]\(\sqrt{40}\)[/tex]