Rational and Irrational Numbers exercise - Free Printable

To solve this problem, we need to sort the numbers into two groups: Rational and Irrational.

Here is the rule to remember:
* Rational Numbers: Can be written as a fraction. This includes whole numbers, terminating decimals (decimals that end), and repeating decimals (decimals with a pattern that repeats forever). Also, square roots or cube roots of "perfect" numbers are rational.
* Irrational Numbers: Cannot be written as a simple fraction. Their decimal forms go on forever without any repeating pattern. Famous examples are $\pi$ and $e$. Also, square roots or cube roots of non-perfect numbers are irrational.

Let's look at each number one by step:

1. 3.14: This is a decimal that ends. It can be written as a fraction ($\frac{314}{100}$). -> Rational
2. 0.444: This is a decimal that ends. It can be written as a fraction ($\frac{444}{1000}$). -> Rational
3. $\sqrt{121}$: We need to find the square root. $11 \times 11 = 121$. So, $\sqrt{121} = 11$. Since 11 is a whole number, it is rational. -> Rational
4. $\sqrt{1000}$: Is there a whole number that you can multiply by itself to get exactly 1000? No ($31 \times 31 = 961$ and $32 \times 32 = 1024$). So this is an irrational number. -> Irrational
5. $\pi$: Pi is a famous irrational number. It never ends and never repeats. -> Irrational
6. $\frac{\pi}{7}$: Even though it is divided by 7, $\pi$ is still irrational. An irrational number divided by a rational number is still irrational. -> Irrational
7. $0.333...$: The dots mean it goes on forever. The digit 3 repeats. Repeating decimals are rational. -> Rational
8. $1.12313...$: The dots mean it goes on forever. There is no repeating pattern shown (like 123123). Non-repeating infinite decimals are irrational. -> Irrational
9. $0.\overline{12}$: The bar over the 12 means "12" repeats forever ($0.121212...$). Repeating decimals are rational. -> Rational
10. $e$: Euler's number ($e$) is a famous irrational number, just like $\pi$. -> Irrational
11. $\sqrt[3]{9}$: Is there a whole number multiplied by itself three times that equals 9? No ($2 \times 2 \times 2 = 8$ and $3 \times 3 \times 3 = 27$). So this is irrational. -> Irrational
12. $2.\overline{9}$: The bar means the 9 repeats forever. Repeating decimals are rational. -> Rational
13. $2e$: This is 2 times $e$. Since $e$ is irrational, multiplying it by 2 keeps it irrational. -> Irrational
14. $1.\overline{234}$: The bar means "234" repeats forever. Repeating decimals are rational. -> Rational
15. $\sqrt[3]{8}$: We need the cube root. $2 \times 2 \times 2 = 8$. So, $\sqrt[3]{8} = 2$. Since 2 is a whole number, it is rational. -> Rational

Now, let's group them for the final answer.

Final Answer:
Rational:
* 3.14
* 0.444
* $\sqrt{121}$
* $0.333...$
* $0.\overline{12}$
* $2.\overline{9}$
* $1.\overline{234}$
* $\sqrt[3]{8}$

Irrational:
* $\sqrt{1000}$
* $\pi$
* $\frac{\pi}{7}$
* $1.12313...$
* $e$
* $\sqrt[3]{9}$
* $2e$