Question

Prove that \[2 - 3\sqrt 5 \]is irrational?

Answer 1

Hint: Irrational numbers are the numbers which are not terminating in nature and we cannot find the exact values of those number, their decimal values are either repeating and non terminating, or non repeating and non terminating, example of irrational number can be seen as value of pie.

Complete step-by-step answer:
 Here we know the definition of irrational numbers and thus have to show that the given expression value is non terminating and we can not find the exact value after solving the expression.
Here we know that “two” is the exact value and is rational, now “three” is also a rational number hence to prove this expression as irrational we need to show that root five is a irrational number, on solving we get:
\[
   \Rightarrow value\,of\,root\,5 \\
   \Rightarrow \sqrt 5 = 2.2360..... \;
 \]
Here the value of root five is non terminating hence the solution of the given expression will lead to irrational numbers.
Here when root five will multiply with three then again it will give an irrational number, then when this number is subtracted by two then again the solution will be irrational.

Note: Here we know the property of irrational number that the values which are non repeating and non terminating will said to be irrational, hence to solve this sort of question we need to find the rational and irrational numbers, then simply we can reach to the solution that the given expression will be rational or irrational on solving.