choose all of the rational numbers. $\frac{-5}{0}$ $\frac{0}{0}$ $0.11overline{1}$ $\frac{0}{-10}$ $sqrt{2}$…

Explanation:

Step1: Recall the definition of rational numbers

A rational number is a number that can be written as $\frac{p}{q}$ where $p,q$ are integers and $q\neq0$.

Step2: Analyze $\frac{- 5}{0}$ and $\frac{0}{0}$

Division by zero is undefined, so $\frac{-5}{0}$ and $\frac{0}{0}$ are not rational numbers.

Step3: Analyze $0.11\overline{1}$

Let $x = 0.11\overline{1}$. Then $10x=1.11\overline{1}$ and $10x - x=1.11\overline{1}-0.11\overline{1}$, $9x = 1$, $x=\frac{1}{9}$, so it is rational.

Step4: Analyze $\frac{0}{-10}$

Since $\frac{0}{-10}=0$ and $0=\frac{0}{1}$ (where $0$ and $1$ are integers), it is rational.

Step5: Analyze $\sqrt{2}$

$\sqrt{2}$ is an irrational number as it cannot be written as a fraction of two integers.

Step6: Analyze $\frac{2}{-2}$

$\frac{2}{-2}=-1$ and $-1=\frac{-1}{1}$, so it is rational.

Step7: Analyze $\sqrt{36}$

$\sqrt{36} = 6$ and $6=\frac{6}{1}$, so it is rational.

Answer:

$0.11\overline{1},\frac{0}{-10},\frac{2}{-2},\sqrt{36}$