Question

Which of the following is irrational?(1 point) Responses 3.6363636363636363... 3.6363636363636363... −7837 negative 7 Start Fraction 8 over 37 End Fraction 52.781654292 52.781654292 3√4 Start Fraction Start Root 3 End Root over 4 End Fraction

Answer 1

To determine which number is irrational from the provided options, let's analyze each one:

1. **3.6363636363636363...**: This is a repeating decimal, which can be expressed as the fraction \( \frac{363636}{100000} \) or simplified, making it rational.

2. **3.6363636363636363... − 7837**: Since \( 3.6363636363636363... \) is rational and subtracting a rational number (−7837) from it results in a rational number, this is also rational.

3. **−7**: This is an integer, which is also a rational number.

4. **\( \frac{8}{37} \)**: This is a fraction of two integers, so it is rational.

5. **52.781654292**: This is a terminating decimal, which can be expressed as a fraction, hence it is rational.

6. **\( 3 \sqrt{4} \)**: This simplifies to \( 3 \times 2 = 6 \), which is also rational.

7. **\( \frac{\sqrt{3}}{4} \)**: While both parts of the fraction are real numbers, since \( \sqrt{3} \) is irrational, the entire expression \( \frac{\sqrt{3}}{4} \) is irrational because it cannot be expressed as a fraction of two integers.

Based on the above analysis, the only irrational number from the options is **\( \frac{\sqrt{3}}{4} \)**.