`""^(L)sqrtM` is surd of order……, where M is a rational number, L is a positive integer and `""^(L)sqrtM` is irrational.

Options

A. L
B. M
C. 2
D. 4

Answer

To determine the order of the surd \( \sqrt[L]{M} \), where \( M \) is a rational number and \( L \) is a positive integer, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Expression**:
The expression \( \sqrt[L]{M} \) can be rewritten using exponent notation. The \( L \)-th root of \( M \) can be expressed as:
\[ \sqrt[L]{M} = M^{\frac{1}{L}} \]

2. **Identifying the Components**:
Here, \( M \) is given to be a rational number, and \( L \) is a positive integer.

3. **Analyzing the Irrationality**:
We are told that \( \sqrt[L]{M} \) is irrational. This means that \( M^{\frac{1}{L}} \) must not simplify to a rational number.

4. **Condition for Irrationality**:
For \( M^{\frac{1}{L}} \) to be irrational, \( M \) must not be a perfect \( L \)-th power of a rational number. In other words, if \( M = a^L \) for some rational \( a \), then \( \sqrt[L]{M} \) would be rational.

5. **Conclusion on the Order of the Surd**:
Since \( L \) is the index of the root, we conclude that the order of the surd \( \sqrt[L]{M} \) is \( L \).

### Final Answer:
The order of the surd \( \sqrt[L]{M} \) is \( L \).

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