To determine which of the following numbers are irrational, let's analyze each one carefully:
1. [tex]\(3.6363636363636363 \ldots\)[/tex]:
This number appears to be a repeating decimal, specifically 3.6363... (with 63 repeating). Repeating decimals are always rational because they can be expressed as a fraction (a ratio of two integers). For example, the repeating decimal 0.6363... can be written as a fraction. Therefore, 3.6363636363636363... is rational.
2. [tex]\(\frac{\sqrt{3}}{4}\)[/tex]:
To determine if this number is rational, consider the value of [tex]\(\sqrt{3}\)[/tex]. The square root of 3 is an irrational number, meaning it cannot be expressed as a fraction of two integers. When you divide an irrational number by a rational number (other than zero), the result is still irrational. Therefore, [tex]\(\frac{\sqrt{3}}{4}\)[/tex] is irrational.
3. 52.781654292:
This number is given to a finite number of decimal places (meaning it terminates after a certain number of digits). Since it has a finite number of digits after the decimal point, it can be expressed as a fraction. Thus, 52.781654292 is a rational number.
4. [tex]\(-7 + \frac{8}{37}\)[/tex]:
This number is a sum of a rational number (-7) and a fraction (8/37), where both -7 and 8/37 are rational. The sum or difference of two rational numbers is always rational. Therefore, [tex]\(-7 + \frac{8}{37}\)[/tex] is rational.
Conclusively, the only irrational number among the given options is [tex]\(\frac{\sqrt{3}}{4}\)[/tex].
