Question

Find the sum of 2/3 and 5/12 in simplest form.

Answer 1

`12\sqrt[]{3}`

"The result is irrational because it cannot be written as the ratio of two integers and its decimal expansion does not terminate or repeat.

1) Note that we have here two irrational numbers:

`2\sqrt[]{3}+5\sqrt[]{12}`

2) We can simplify a little bit more, the square root of 12 like this:

`\begin{gathered} 5\sqrt[]{12} \\ 5\sqrt[]{2^2\cdot3} \\ 5\cdot2\sqrt[]{3} \\ 10\sqrt[]{3} \end{gathered}`

Note that now, we have two radicals with the same radicand. So we can proceed with that:

`\begin{gathered} 2\sqrt[]{3}+5\sqrt[]{12} \\ 2\sqrt[]{3}+10\sqrt[]{3} \\ 12\sqrt[]{3} \end{gathered}`

3) We can now answer the 2nd part.

"The result is irrational because it cannot be written as the ratio of two integers and its decimal expansion does not terminate or repeat.