Question

How can I find out how each possible answer is rational or irrational…can you show me step by step?

Answer 1

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.

`\begin{gathered} n=(p)/(q) \\ p,q\in Z \end{gathered}`

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:

`(5)/(6)+\frac{\sqrt[]{12}}{3}`

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.

`(5)/(6)+\frac{\sqrt[]{12}}{3}=(5)/(6)+\frac{2*\sqrt[]{12}}{2*3}=(5)/(6)+\frac{2\sqrt[]{12}}{6}=\frac{5+2\sqrt[]{12}}{6}`

We can also simplify the square root of twelve

`\frac{5+2\sqrt[]{12}}{6}=\frac{5+2\sqrt[]{3*4}}{6}=\frac{5+2\sqrt[]{3}\sqrt[]{4}}{6}=\frac{5+4\sqrt[]{3}}{6}`

This is the simplest form of writing the number as a single fraction. Now, let's analyze the numerator and denominator.

`6\in Z,(5+4\sqrt[]{3})\\otin Z`

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

`(5)/(6)-\frac{\sqrt[]{36}}{3}`

Again, let's make the denominators equal:

`(5)/(6)-\frac{\sqrt[]{36}}{3}=(5)/(6)-\frac{2*\sqrt[]{36}}{2*3}=(5)/(6)-\frac{2\sqrt[]{36}}{6}=\frac{5-2\sqrt[]{36}}{6}`

36 is a perfect square, and its square root is 6.

`\frac{5-2\sqrt[]{36}}{6}=(5-2*6)/(6)=(5-12)/(6)=(-7)/(6)`

Since - 7 and 6 are both integers, this number is rational.

The number for item C:

`(5)/(6)\frac{\sqrt[]{12}}{3}`

When we want to do a product between two fractions, we just multiply denominator by denominator, and numerator by numerator:

`(5)/(6)\frac{\sqrt[]{12}}{3}=\frac{5*\sqrt[]{12}}{6*3}=\frac{5\sqrt[]{12}}{18}`

We can simplify this expression the same way we did on item A.

`\frac{5\sqrt[]{12}}{18}=\frac{10\sqrt[]{3}}{18}=\frac{5\sqrt[]{3}}{9}`

Since the numerator is not an integer

`5\sqrt[]{3}\\otin Z`

This number is not rational.

For the last item, D, we have:

`(5)/(6)\colon\frac{\sqrt[]{36}}{3}`

Dividing a fraction by other, is the same as multiplying the first one by the inverse of the other.

`(5)/(6)\colon\frac{\sqrt[]{36}}{3}=(5)/(6)*\frac{3}{\sqrt[]{36}}=\frac{15}{6\sqrt[]{36}}=(15)/(36)=(5)/(12)`

Since both 5 and 12 are integers, this number is rational.