Which one of the following numbers is irrational (that is, not rational)?

Introduction / Context:
This question examines your understanding of rational and irrational numbers. A rational number can be written as a ratio of two integers, whereas an irrational number cannot be expressed in such a form and has a non terminating, non repeating decimal expansion. You are given several expressions involving square roots and asked to identify which one is irrational.

Given Data / Assumptions:

- Option A: 3√8. - Option B: 3√64. - Option C: √64. - Option D: 24. - Standard definitions of rational and irrational numbers apply.

Concept / Approach:
The key concept is that square roots of perfect squares are integers and therefore rational. Square roots of non perfect squares are irrational. Multiplying an irrational number by a non zero rational number still gives an irrational number. We evaluate each option by simplifying the square roots where possible and then classifying the result as rational or irrational based on these rules.

Step-by-Step Solution:
Step 1: Consider √64. Since 64 is a perfect square (8 * 8 = 64), √64 = 8, which is an integer and therefore rational. Step 2: Consider 3√64. We already know √64 = 8, so 3√64 = 3 * 8 = 24, which is also an integer and rational. Step 3: Option D is 24 explicitly, which is clearly a rational integer. Step 4: Now consider 3√8. Here the radicand 8 is not a perfect square. We can write √8 as √(4 * 2) = √4 * √2 = 2√2. Step 5: Then 3√8 = 3 * 2√2 = 6√2. Step 6: The number √2 is known to be irrational, and multiplying it by the non zero rational number 6 keeps the product irrational. Step 7: Therefore 3√8 (equal to 6√2) is irrational, while all other options simplify to rational numbers.

Verification / Alternative check:
You can further verify by approximating the values. √2 is about 1.414..., so 6√2 is about 8.485..., which has a non terminating and non repeating decimal expansion, consistent with being irrational. In contrast, √64 is exactly 8, 3√64 is exactly 24 and 24 is clearly a whole number; all of these have finite decimal expansions and are rational. This confirms the classification derived from simplifying the square roots.

Why Other Options Are Wrong:
Option B, 3√64, equals 24 and is rational. Option C, √64, equals 8 and is also rational. Option D is simply 24, a rational integer. None of these can be categorized as irrational. Only option A involves the square root of a non perfect square and remains irrational after simplification.

Common Pitfalls:
Some learners may think that any expression with a square root is automatically irrational, which is not true for perfect squares like 64. Others may forget to simplify radicals like √8 and fail to recognize that 3√8 contains √2 as a factor. Remember that the critical factor is whether the number under the root is a perfect square, not just the presence of the radical sign.

Final Answer:
The irrational number among the given options is 3√8, which corresponds to option A.