Understanding Rational and Irrational Numbers

In mathematics, numbers can be classified into different types, including rational and irrational numbers. 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 5, which is $\frac{5}{1}$), fractions (like $\frac{1}{2}$), and terminating or repeating decimals (like 0.75 or 0.333...).

An irrational number is a number that cannot be expressed as a simple fraction $\frac{p}{q}$. When written as a decimal, irrational numbers are non-terminating and non-repeating. Famous examples include $\pi$ and $\sqrt{2}$.

The question asks us to identify which of the given numbers is NOT rational, meaning we are looking for the irrational number among the options.

Evaluating Each Number Option

Let's examine each number provided in the options:

Option 1: Evaluating $\sqrt{64}$

This is the square root of 64. We need to find a number that, when multiplied by itself, equals 64.

Calculation:

$\sqrt{64} = 8$

Explanation: The number 8 is an integer. Any integer can be written as a fraction with a denominator of 1 (for example, $8 = \frac{8}{1}$). Since 8 can be expressed in the form $\frac{p}{q}$ where $p=8$ and $q=1$ (both integers, $q \neq 0$), 8 is a rational number.

Option 2: Evaluating $\sqrt[3]{64}$

This is the cube root of 64. We need to find a number that, when multiplied by itself three times, equals 64.

Calculation:

$\sqrt[3]{64} = 4$

Explanation: The number 4 is an integer. Similar to the previous option, 4 can be written as a fraction ($\frac{4}{1}$). Since 4 can be expressed in the form $\frac{p}{q}$ where $p=4$ and $q=1$ (both integers, $q \neq 0$), 4 is a rational number.

Option 3: Evaluating $\sqrt[3]{8}$

This is the cube root of 8. We need to find a number that, when multiplied by itself three times, equals 8.

Calculation:

$\sqrt[3]{8} = 2$

Explanation: The number 2 is an integer. 2 can be written as a fraction ($\frac{2}{1}$). Since 2 can be expressed in the form $\frac{p}{q}$ where $p=2$ and $q=1$ (both integers, $q \neq 0$), 2 is a rational number.

Option 4: Evaluating $\sqrt{8}$

This is the square root of 8. We need to find a number that, when multiplied by itself, equals 8.

Calculation:

$\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$

Explanation: The number $\sqrt{2}$ is a well-known irrational number. It cannot be expressed as a fraction of two integers, and its decimal representation is non-terminating and non-repeating ($\sqrt{2} \approx 1.41421356...$). When an irrational number is multiplied by a non-zero rational number (like 2 in this case), the result is also an irrational number. Therefore, $2\sqrt{2}$ (or $\sqrt{8}$) is an irrational number.

Conclusion: Identifying the Non-Rational Number

Based on our evaluations:

• $\sqrt{64} = 8$, which is rational.
• $\sqrt[3]{64} = 4$, which is rational.
• $\sqrt[3]{8} = 2$, which is rational.
• $\sqrt{8} = 2\sqrt{2}$, which is irrational.

The number that is NOT rational among the given options is $\sqrt{8}$.

Revision Table: Rational vs. Irrational

Additional Information on Number Types

Understanding number classifications is fundamental in mathematics. Here are a few key points:

• Integers: Whole numbers, both positive and negative, including zero ($\dots, -2, -1, 0, 1, 2, \dots$). All integers are rational numbers.
• Natural Numbers: Positive integers, sometimes starting from 0 ($1, 2, 3, \dots$ or $0, 1, 2, 3, \dots$). Natural numbers are a subset of integers.
• Real Numbers: This set includes all rational and irrational numbers. Every number on the number line is a real number.
• The product of a non-zero rational number and an irrational number is always irrational.
• The sum or difference of a rational number and an irrational number is always irrational.
• The sum, difference, product, or quotient of two rational numbers is always rational (provided the denominator is not zero for division).
• The sum, difference, product, or quotient of two irrational numbers can be either rational or irrational. For example, $\sqrt{2} + (-\sqrt{2}) = 0$ (rational), but $\sqrt{2} + \sqrt{2} = 2\sqrt{2}$ (irrational). Also, $\sqrt{2} \times \sqrt{2} = 2$ (rational), but $\sqrt{2} \times \sqrt{3} = \sqrt{6}$ (irrational).

Identifying whether a square root $\sqrt{n}$ or cube root $\sqrt[3]{n}$ is rational depends on whether $n$ is a perfect square or a perfect cube, respectively.

• $\sqrt{n}$ is rational if and only if $n$ is a perfect square (e.g., 1, 4, 9, 16, 25, 64...).
• $\sqrt[3]{n}$ is rational if and only if $n$ is a perfect cube (e.g., 1, 8, 27, 64, 125...).

In this question, 64 is both a perfect square ($8^2$) and a perfect cube ($4^3$), while 8 is a perfect cube ($2^3$) but not a perfect square. This explains why $\sqrt{64}$, $\sqrt[3]{64}$, and $\sqrt[3]{8}$ are rational, but $\sqrt{8}$ is not.