`2^(sqrt3)` is :

Options

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

Answer

To determine whether \( 2^{\sqrt{3}} \) is a rational or irrational number, we can follow these steps:

### Step 1: Understand the components of the expression
- The base of the expression is \( 2 \), which is a rational number.
- The exponent is \( \sqrt{3} \), which is an irrational number.

**Hint:** Identify the nature (rational or irrational) of both the base and the exponent.

### Step 2: Recall the properties of exponents
- A number raised to a rational exponent is rational if the exponent is a whole number.
- A rational number raised to an irrational exponent results in an irrational number.

**Hint:** Remember the rules regarding rational and irrational numbers when used as bases and exponents.

### Step 3: Analyze the expression \( 2^{\sqrt{3}} \)
- Since \( 2 \) is rational and \( \sqrt{3} \) is irrational, we apply the property that a rational number raised to an irrational power yields an irrational number.

**Hint:** Think about how the properties of numbers interact when combined in mathematical operations.

### Step 4: Conclude the nature of \( 2^{\sqrt{3}} \)
- Therefore, \( 2^{\sqrt{3}} \) is an irrational number.

**Hint:** Summarize your findings based on the properties discussed.

### Final Answer:
\( 2^{\sqrt{3}} \) is an irrational number.