2/3 is a rational number whereas `sqrt2/sqrte` is

Options

A. also a rational number
B. an irrational number
C. not a number
D. a natural periodic number

Answer

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

### Step 1: Understand Rational and Irrational Numbers
A rational number can be expressed in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). An irrational number cannot be expressed in this form.

**Hint:** Remember that rational numbers can be written as fractions, while irrational numbers cannot.

### Step 2: Analyze the Components
We have \( \sqrt{2} \) and \( \sqrt{3} \). We need to determine if these square roots are rational or irrational.

**Hint:** Check if the numbers under the square roots are perfect squares.

### Step 3: Determine the Nature of \( \sqrt{2} \) and \( \sqrt{3} \)
- \( \sqrt{2} \) is not a perfect square, and it cannot be expressed as a fraction of two integers. Therefore, \( \sqrt{2} \) is an irrational number.
- Similarly, \( \sqrt{3} \) is also not a perfect square and cannot be expressed as a fraction of two integers. Thus, \( \sqrt{3} \) is also an irrational number.

**Hint:** Recall that square roots of non-perfect squares are always irrational.

### Step 4: Analyze the Fraction \( \frac{\sqrt{2}}{\sqrt{3}} \)
Now we consider the fraction \( \frac{\sqrt{2}}{\sqrt{3}} \). Since both the numerator and the denominator are irrational, we need to check if their ratio can be rational.

**Hint:** Think about the properties of the division of irrational numbers.

### Step 5: Conclusion
The ratio of two irrational numbers can be either rational or irrational. However, in this case, \( \frac{\sqrt{2}}{\sqrt{3}} \) does not simplify to a fraction of two integers. Therefore, it remains irrational.

Thus, we conclude that \( \frac{\sqrt{2}}{\sqrt{3}} \) is an irrational number.

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