Question

Which of the following is irrational?(1 point)
Responses

52.781654292
52.781654292

−7837
negative 7 Start Fraction 8 over 37 End Fraction

3.6363636363636363...
3.6363636363636363...

3√4

Answer 1

To determine which of the given numbers is irrational, we need to understand the definitions:

- A rational number is a number that can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b \neq 0 \).
- An irrational number cannot be expressed as a simple fraction, and it has non-repeating, non-terminating decimal expansions.

Now let's analyze each option:

1. **52.781654292** - This is a terminating decimal. Therefore, it is a rational number.

2. **−7837** - This is an integer. All integers are rational numbers, so this is rational.

3. **3.6363636363636363...** - This is a repeating decimal (specifically, it can be represented as \( 3.636363... = \frac{109}{30} \)). Therefore, it is a rational number.

4. **3√4** - This is the cube root of 4. The value of \( \sqrt{4} = 2 \), and the cube root of 2, \( \sqrt[3]{2} \), is known to be an irrational number.

Based on the above analysis, the only irrational number in the list is **3√4**.