Question

select all numbers that are irrational.

\\(\sqrt{0.156}\\)

\\(368.5468432\dots\\)

\\(14.19274128\dots\\)

\\(253,897,615.\bar{3}\\)

\\(77.65488319\\)

\\(\sqrt{\frac{4}{9}}\\)

Explanation:

Step1: Analyze the first number

$$\sqrt{0.156} = \sqrt{\frac{156}{1000}}$$

Since \(156\) and \(1000\) are not perfect squares, this is irrational.

Step2: Analyze the second number

$$368.5468432\dots$$

This is a non-terminating, non-repeating decimal, which is irrational.

Step3: Analyze the third number

$$14.19274128\dots$$

This is a non-terminating, non-repeating decimal, which is irrational.

Step4: Analyze the fourth number

$$253,897,615.\overline{3}$$

This is a repeating decimal, which is rational.

Step5: Analyze the fifth number

$$77.65488319$$

This is a terminating decimal, which is rational.

Step6: Analyze the sixth number

$$\sqrt{\frac{4}{9}} = \frac{2}{3}$$

This simplifies to a fraction of integers, which is rational.

Answer:

• \(\sqrt{0.156}\)
• \(368.5468432\dots\)
• \(14.19274128\dots\)