`2sqrt(3/x)` is :

Options

A. a natural number
B. an integer number
C. a rational number
D. an irrational number

Answer

To determine the nature of the expression \(2\sqrt{\frac{3}{x}}\), we will analyze it step by step.

### Step 1: Understanding the Expression
The expression is \(2\sqrt{\frac{3}{x}}\). Here, we have a constant \(2\) multiplied by the square root of the fraction \(\frac{3}{x}\).

**Hint:** Identify the components of the expression, such as constants and variables.

### Step 2: Analyzing the Square Root
The square root function, \(\sqrt{y}\), is defined for non-negative values of \(y\). Therefore, we need to ensure that \(\frac{3}{x} \geq 0\). This means:
- \(x\) must be positive (since \(3\) is positive).

**Hint:** Consider the conditions under which the square root is defined.

### Step 3: Nature of \(\sqrt{3}\)
The value \(\sqrt{3}\) is an irrational number because it cannot be expressed as a fraction of two integers.

**Hint:** Recall that numbers like \(\sqrt{2}\), \(\sqrt{3}\), and \(\sqrt{5}\) are examples of irrational numbers.

### Step 4: Multiplying by 2
Now, we multiply the irrational number \(\sqrt{3}\) by the integer \(2\):
\[ 2\sqrt{3} \]
The product of a rational number (in this case, \(2\)) and an irrational number (\(\sqrt{3}\)) is still irrational.

**Hint:** Remember that multiplying or dividing an irrational number by a non-zero rational number results in an irrational number.

### Step 5: Considering the Variable \(x\)
Since \(x\) is in the denominator, we must also consider its effect. If \(x\) is a positive number, \(\frac{3}{x}\) remains positive, and thus \(\sqrt{\frac{3}{x}}\) is defined and remains irrational.

**Hint:** Ensure that \(x\) does not equal zero, as that would make the expression undefined.

### Conclusion
Since \(2\sqrt{\frac{3}{x}}\) is the product of a rational number and an irrational number, it is irrational as long as \(x\) is positive. Therefore, the expression \(2\sqrt{\frac{3}{x}}\) is an **irrational number**.

**Final Answer:** The expression \(2\sqrt{\frac{3}{x}}\) is an irrational number (Option 4).