Which of the following number is irrational?

Options

A. `sqrt((4)/(9))`
B. `(sqrt(1250))/(sqrt(8))`
C. `(sqrt(8))`
D. `(sqrt(24))/(sqrt(6))`

Answer

To determine which of the following numbers is irrational, we will analyze each option step by step.

### Step 1: Identify the options
Let's assume the options provided are:
1. \( \sqrt{4} \)
2. \( \frac{\sqrt{4}}{3} \)
3. \( \sqrt{1250} \)
4. \( \frac{\sqrt{24}}{\sqrt{6}} \)

### Step 2: Analyze each option

**Option 1: \( \sqrt{4} \)**
- The square root of 4 is 2, which is a rational number.

**Conclusion:** \( \sqrt{4} = 2 \) (Rational)

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**Option 2: \( \frac{\sqrt{4}}{3} \)**
- Since \( \sqrt{4} = 2 \), we have \( \frac{2}{3} \), which is also a rational number.

**Conclusion:** \( \frac{\sqrt{4}}{3} = \frac{2}{3} \) (Rational)

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**Option 3: \( \sqrt{1250} \)**
- We can simplify \( \sqrt{1250} \):
- Factor 1250: \( 1250 = 25 \times 50 = 25 \times 25 \times 2 = 5^3 \times 2 \).
- Thus, \( \sqrt{1250} = \sqrt{5^3 \times 2} = 5\sqrt{10} \).
- Since \( \sqrt{10} \) is not a perfect square, it is irrational.

**Conclusion:** \( \sqrt{1250} = 5\sqrt{10} \) (Irrational)

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**Option 4: \( \frac{\sqrt{24}}{\sqrt{6}} \)**
- Simplifying \( \frac{\sqrt{24}}{\sqrt{6}} \):
- \( \sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6} \).
- Therefore, \( \frac{\sqrt{24}}{\sqrt{6}} = \frac{2\sqrt{6}}{\sqrt{6}} = 2 \).
- Since 2 is a rational number, this option is rational.

**Conclusion:** \( \frac{\sqrt{24}}{\sqrt{6}} = 2 \) (Rational)

### Final Answer:
The only irrational number among the options is **Option 3: \( \sqrt{1250} \)**.

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