Definition check (surd order): L√M denotes the L-th root of M. If M is rational, L is a positive integer, and L√M is irrational, then the surd is of what order?

Introduction / Context:
A surd is an irrational root expression. The “order” refers to the index of the radical (the number indicating which root). For L√M, L is the order (e.g., cube root has order 3).

Given Data / Assumptions:

• Expression form: L√M, i.e., the L-th root of M.
• M is rational; L ∈ positive integers.
• L√M is irrational (hence a surd).

Concept / Approach:
By definition, in n√A the order (or index) is n. The irrationality condition just certifies it qualifies as a surd; it does not change the order.

Step-by-Step Solution:
Identify index: LTherefore, the surd is of order L

Verification / Alternative check:
Examples: √2 (order 2), 3√5 (order 3), 5√7 (order 5). The order equals the radical’s index.

Why Other Options Are Wrong:
M is the radicand, not the order; 2 or 4 are specific indices, not general; L/2 is unrelated.

Common Pitfalls:
Confusing the radicand (inside the root) with the index (outside, as the small number on the radical sign).

Final Answer:
L