To determine which of the numbers given is irrational, we need to analyze each one.
-
-7 8/37: This number is a mixed number, which can be converted to an improper fraction. The whole part is -7, and the fractional part is \( \frac{8}{37} \). Therefore, it can be expressed as: \[ -7 + \frac{8}{37} = \frac{-259 + 8}{37} = \frac{-251}{37} \] Since both the numerator and denominator are integers, this number is rational.
-
3.6363636363636363: This number appears to be a repeating decimal (3.6363...) that can be expressed as a fraction. The repeating part (36) translates to a rational number since all repeating decimals can be represented as a fraction. Thus, this number is rational.
-
3√4 (or \( \sqrt[3]{4} \)): The cube root of 4 is not a perfect cube, and it cannot be expressed as a fraction of integers. The cube root of a non-perfect cube is irrational. Thus, \( \sqrt[3]{4} \) is irrational.
-
52.781654292: This number is a decimal with a finite number of digits after the decimal point. All finite decimals can be expressed as fractions and are therefore rational.
From the analysis, the only irrational number among the given options is:
3√4 (or \( \sqrt[3]{4} \)).