Question

Here are two expressions with a mixing number:

Answer 1

An irrational number is a real number that can't be written as a fraction: since all rational numbers ($\mathbb{Q}$) are real numbers ($\mathbb{R}$), the irrational numbers are all elements in $\mathbb{R}$ that don't belong to $\mathbb{Q}$.

The rational numbers are closed with respect to sum and multiplication, which means that the sum/product of two rational numbers is a rational number.

Note that, of all the options, only $\sqrt{3}$ is irrational, because the other options are

$3 = \dfrac{3}{1},\quad 2 =\dfrac{2}{1},\quad 0.\overline{23} = \dfrac{23}{99}$

so they all can be written as fraction. This means that

$-\dfrac{5}{9} \times 3,\quad -\dfrac{5}{9} \times 2,\quad -\dfrac{5}{9} \times\dfrac{23}{99}$

are all rational numbers, because they are the multiplication of two rational numbers.

So, your only hope to get an irrational number is to multiply

$-\dfrac{5}{9}\times \sqrt{3}$

Expression 2 works exactly in the same way, because everything we said about the product is true for the sum as well.