Question

classify the numbers as rational or irrational. $sqrt{31}$, $4$, $sqrt{31} cdot 4$
rational
irrational

Explanation:

Step1: Define rational numbers

A rational number can be written as $\frac{p}{q}$, where $p,q$ are integers, $q
eq0$. Integers are rational.

Step2: Classify 4

4 is an integer, so it is rational.

Step3: Classify $\sqrt{31}$

31 is not a perfect square, so $\sqrt{31}$ cannot be written as $\frac{p}{q}$ (integer $p,q$). It is irrational.

Step4: Classify $\sqrt{31} \cdot 4$

Product of non-zero rational and irrational is irrational. $4$ is rational, $\sqrt{31}$ is irrational, so $4\sqrt{31}$ is irrational.

Answer:

rational: $4$
irrational: $\sqrt{31}$, $\sqrt{31} \cdot 4$