Which of the following is irrational?

Question

options:

3.6363636363636363...

3√ over 4

−7 and 8 over 37

52.781654292

Answer 1

To determine which of the given options is irrational, let's analyze each option:

1. 3.6363636363636363...: This number is a repeating decimal (the "63" repeats indefinitely), which can be expressed as a fraction. Any repeating decimal is rational. Therefore, this number is rational.

2. 3√ over 4 (or $\sqrt[3]{4}$): The cube root of a rational number (like 4) is also rational. Therefore, $\sqrt[3]{4}$ is rational.

3. −7 and 8 over 37 (or $-7+\frac{8}{37}$): The expression can be simplified as a fraction after doing the arithmetic. Since it is a simple arithmetic operation involving whole numbers and a fraction, it is rational.

4. 52.781654292: This is a decimal that terminates and can be expressed as a fraction. Therefore, it is rational.

Among all the options, none of them are irrational. However, if you're looking for irrational numbers, it's important to know examples like $\pi$, $\sqrt{2}$, and other non-repeating, non-terminating decimals.

Thus, none of the given options are irrational.