Identifying Rational Numbers from Cube Roots

Understanding the difference between rational and irrational numbers is key to solving this problem. A rational number is any number that can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q$ is not zero. Examples include integers like 2 (which is $\frac{2}{1}$), fractions like $\frac{1}{2}$, and terminating or repeating decimals like 0.5 or 0.333...

An irrational number is a number that cannot be expressed as a simple fraction. Its decimal representation is non-terminating and non-repeating. Famous examples include $\pi$ and $\sqrt{2}$.

The question asks us to identify which of the given cube roots is a rational number. Let's evaluate each option:

Evaluating the Cube Root Options

1. $\sqrt[3]{2}$: This represents the number that, when multiplied by itself three times, equals 2. Since 2 is not a perfect cube (no integer multiplied by itself three times equals 2), $\sqrt[3]{2}$ is an irrational number. Its decimal form is non-terminating and non-repeating.
2. $\sqrt[3]{8}$: This represents the number that, when multiplied by itself three times, equals 8. We know that $2 \times 2 \times 2 = 8$. Therefore, $\sqrt[3]{8} = 2$. The number 2 can be expressed as the fraction $\frac{2}{1}$, where 2 and 1 are integers and the denominator is not zero. Thus, 2 is a rational number.
3. $\sqrt[3]{4}$: This represents the number that, when multiplied by itself three times, equals 4. Since 4 is not a perfect cube, $\sqrt[3]{4}$ is an irrational number.
4. $\sqrt[3]{12}$: This represents the number that, when multiplied by itself three times, equals 12. Since 12 is not a perfect cube, $\sqrt[3]{12}$ is an irrational number.

Conclusion: Identifying the Rational Number

Based on our evaluations, only $\sqrt[3]{8}$ simplifies to an integer, which is a rational number. The other options, $\sqrt[3]{2}$, $\sqrt[3]{4}$, and $\sqrt[3]{12}$, are cube roots of numbers that are not perfect cubes, making them irrational numbers.

Revision Table: Rational vs. Irrational Numbers

Additional Information: Perfect Cubes and Cube Roots

A perfect cube is an integer that is the cube of another integer. For example, 8 is a perfect cube because $2^3 = 8$. Similarly, 27 is a perfect cube because $3^3 = 27$.

Finding the cube root of a perfect cube results in an integer. Since all integers are rational numbers, the cube root of a perfect cube is always rational. If a number is not a perfect cube, its cube root is typically an irrational number (unless the original number is 0).