First, to determinate which numbers can be written as a rational or irrational, first we need to define those two terms.
A rational number is any real number that can be expressed as the quotient of two integers.
[tex]\begin{gathered} n=\frac{p}{q} \\ p,q\in Z \end{gathered}[/tex]n is a rational number.
A irrational number is any real number that cannot be expressed as the quotient of two integers.
Now that we know what a rational number is, let's check the items in the question.
For the item A, we have:
[tex]\frac{5}{6}+\frac{\sqrt[]{12}}{3}[/tex]To simplify this expression, first let's multiply both numerator and denominator of the second term by 2, so we can make the denominators of both terms equal.
[tex]\frac{5}{6}+\frac{\sqrt[]{12}}{3}=\frac{5}{6}+\frac{2\times\sqrt[]{12}}{2\times3}=\frac{5}{6}+\frac{2\sqrt[]{12}}{6}=\frac{5+2\sqrt[]{12}}{6}[/tex]We can also simplify the square root of twelve
[tex]\frac{5+2\sqrt[]{12}}{6}=\frac{5+2\sqrt[]{3\times4}}{6}=\frac{5+2\sqrt[]{3}\sqrt[]{4}}{6}=\frac{5+4\sqrt[]{3}}{6}[/tex]This is the simplest form of writing the number as a single fraction. Now, let's analyze the numerator and denominator.
[tex]6\in Z,(5+4\sqrt[]{3})\notin Z[/tex]Since the numerator isn't an integer, then this number is not rational.
Now, let's check the number for item B.
Our number is
[tex]\frac{5}{6}-\frac{\sqrt[]{36}}{3}[/tex]Again, let's make the denominators equal:
[tex]\frac{5}{6}-\frac{\sqrt[]{36}}{3}=\frac{5}{6}-\frac{2\times\sqrt[]{36}}{2\times3}=\frac{5}{6}-\frac{2\sqrt[]{36}}{6}=\frac{5-2\sqrt[]{36}}{6}[/tex]36 is a perfect square, and its square root is 6.
[tex]\frac{5-2\sqrt[]{36}}{6}=\frac{5-2\times6}{6}=\frac{5-12}{6}=\frac{-7}{6}[/tex]Since - 7 and 6 are both integers, this number is rational.
The number for item C:
[tex]\frac{5}{6}\frac{\sqrt[]{12}}{3}[/tex]When we want to do a product between two fractions, we just multiply denominator by denominator, and numerator by numerator:
[tex]\frac{5}{6}\frac{\sqrt[]{12}}{3}=\frac{5\times\sqrt[]{12}}{6\times3}=\frac{5\sqrt[]{12}}{18}[/tex]We can simplify this expression the same way we did on item A.
[tex]\frac{5\sqrt[]{12}}{18}=\frac{10\sqrt[]{3}}{18}=\frac{5\sqrt[]{3}}{9}[/tex]Since the numerator is not an integer
[tex]5\sqrt[]{3}\notin Z[/tex]This number is not rational.
For the last item, D, we have:
[tex]\frac{5}{6}\colon\frac{\sqrt[]{36}}{3}[/tex]Dividing a fraction by other, is the same as multiplying the first one by the inverse of the other.
[tex]\frac{5}{6}\colon\frac{\sqrt[]{36}}{3}=\frac{5}{6}\times\frac{3}{\sqrt[]{36}}=\frac{15}{6\sqrt[]{36}}=\frac{15}{36}=\frac{5}{12}[/tex]Since both 5 and 12 are integers, this number is rational.
