Question

question 4 of 8
select the three correct answers.
carlos is measuring the rotations of a gear in a machine. when he divides the number of rotations by the time in minutes, he gets several decimal values. some of these decimals repeat in a clear pattern, while others do not. select the rational values.
2.overline{136}
5.63
0.12011...
3.14159...
0.overline{72}

Explanation:

Step1: Recall Rational Number Definition

A rational number can be expressed as a fraction $\frac{p}{q}$ (where $q
eq0$), and its decimal either terminates or repeats. Irrational numbers have non - repeating, non - terminating decimals.

Step2: Analyze Each Number

• For \(2.\overline{136}\): The bar over \(136\) indicates that the sequence \(136\) repeats. So, it is a repeating decimal, hence a rational number.
• For \(5.63\): This is a terminating decimal (it has a finite number of decimal places). A terminating decimal can be written as a fraction (e.g., \(5.63=\frac{563}{100}\)), so it is a rational number.
• For \(0.12011\cdots\): The decimal does not show a repeating pattern and seems to be non - terminating and non - repeating. So, it is an irrational number.
• For \(3.14159\cdots\): This is the decimal expansion of \(\pi\), which is a well - known irrational number with a non - repeating, non - terminating decimal expansion.
• For \(0.\overline{72}\): The bar over \(72\) means that \(72\) repeats. So, it is a repeating decimal and thus a rational number.

Answer:

\(2.\overline{136}\), \(5.63\), \(0.\overline{72}\)