Consider the surd inequalities: I. 4√3 > 3√4 and II. 8√2 > 2√8. Which of the following statements correctly describes which inequalities are true?

Introduction / Context:
This question tests comparison of surds by converting them either to decimal approximations or to a common simplified form. Surd comparison is an important skill for quickly checking which of two root based expressions is larger in competitive examinations and numerical reasoning tests.

Given Data / Assumptions:

• Inequality I: 4√3 > 3√4.
• Inequality II: 8√2 > 2√8.
• We must decide whether each inequality is true or false.

Concept / Approach:
One method is to approximate the square roots with decimal values and compare. Another is to simplify expressions by using known roots, such as √4 = 2 and √8 = 2√2, and then compare exactly. Squaring both sides can also help if we are careful to preserve direction when both sides are positive. Here, simplifying using known exact values is efficient and precise.

Step-by-Step Solution:
For inequality I: 4√3 > 3√4. Compute √4 exactly: √4 = 2, so 3√4 = 3 * 2 = 6. So inequality I becomes 4√3 > 6. Approximate √3 ≈ 1.732, so 4√3 ≈ 4 * 1.732 ≈ 6.928. Since 6.928 > 6, inequality I is true. For inequality II: 8√2 > 2√8. Simplify √8 as √(4 * 2) = 2√2, so 2√8 = 2 * 2√2 = 4√2. Thus inequality II becomes 8√2 > 4√2. Both sides have the same surd √2 as a factor, and 8 > 4, so the inequality holds.

Verification / Alternative check:
For a quick decimal check, approximate √2 ≈ 1.414. Then 8√2 ≈ 8 * 1.414 ≈ 11.312 and 4√2 ≈ 4 * 1.414 ≈ 5.656. Clearly 11.312 > 5.656, confirming inequality II. For I, 4√3 ≈ 6.928 and 3√4 = 6 exactly, which confirms I again. Both methods match and show that each inequality is true.

Why Other Options Are Wrong:
Options claiming that only one inequality holds or that neither holds contradict the numeric comparisons above. Since I is true and II is also true, any option that asserts otherwise is incorrect. The phrasing I is false and II is true is directly contradicted by evaluating 4√3 and 3√4.

Common Pitfalls:
A typical mistake is to treat √8 as 8/2 or to mix up properties of roots. Another issue is forgetting that multiplying or dividing by a common positive surd factor such as √2 is allowed when comparing. While decimal approximations are convenient, they should be used with enough precision to avoid rounding errors that could reverse an inequality if values are close.

Final Answer:
Both inequalities are true, so the correct statement is Both I and II.