Question

State whether the number $ (\sqrt 2 - \sqrt 3 )(\sqrt 2 + \sqrt 3 ) $ is rational or irrational. Justify your answer.

Answer 1

Hint: To solve the given problem, we will apply the identity \[\left( {a - b} \right)\left( {a + b} \right){\text{ }} = {\text{ (}}a{{\text{)}}^2} - {(b)^2}\] and then on putting the given values using above identity, we will check whether we get rational or irrational number. Hence, we will get our required answer.

Complete step-by-step answer:
Rational numbers- The rational numbers are numbers which can be expressed as the ratio of two integers, it can be positive numbers, negative numbers or even zero also. Rational numbers are written as \[\dfrac{p}{q},\] where q is not equal to zero.
Irrational numbers - The numbers which are not a rational number are called irrational numbers. Irrational numbers can be written in decimals but cannot be written as the ratio of two integers.
We need to state whether the number $ (\sqrt 2 - \sqrt 3 )(\sqrt 2 + \sqrt 3 ) $ is rational or irrational.
We know that rational numbers can be expressed as the ratio of two integers, while irrational numbers can’t be expressed that way. So, we will check whether we get the ratio of two integers or not.
We also know that \[\left( {a - b} \right)\left( {a + b} \right){\text{ }} = {\text{ (}}a{{\text{)}}^2} - {(b)^2}.\]
So, now on applying the above formula, we get
 $\Rightarrow (\sqrt 2 - \sqrt 3 )(\sqrt 2 + \sqrt 3 ) = {(\sqrt 2 )^2} - {(\sqrt 3 )^2} $
 $
= (2 - 3) \\
= - 1 \\
 $
So, we get $ (\sqrt 2 - \sqrt 3 )(\sqrt 2 + \sqrt 3 ) = - 1. $
We know that \[\left( { - 1} \right)\] is a negative integer, and can be expressed as $ \dfrac{{ - 1}}{1}, $ hence, it is a rational number.
Thus, the number $ (\sqrt 2 - \sqrt 3 )(\sqrt 2 + \sqrt 3 ) $ is a rational number.

Note: The above problem can also be solved by simply multiplying, i.e., with using the identity of algebra, \[\left( {a - b} \right)\left( {a + b} \right){\text{ }} = {\text{ (}}a{{\text{)}}^2} - {(b)^2}.\]
Let us check the alternative method to solve the above problem.
 $ (\sqrt 2 - \sqrt 3 )(\sqrt 2 + \sqrt 3 ) = (\sqrt 2 )(\sqrt 2 ) - (\sqrt 3 )(\sqrt 2 ) + (\sqrt 3 )(\sqrt 2 ) - (\sqrt 3 )(\sqrt 3 ) $
Then on cancelling the negative and positive term in the right-hand side of the equation, we get
 $ (\sqrt 2 - \sqrt 3 )(\sqrt 2 + \sqrt 3 ) = (\sqrt 2 )(\sqrt 2 ) - (\sqrt 3 )(\sqrt 3 ) $
 $ (\sqrt 2 - \sqrt 3 )(\sqrt 2 + \sqrt 3 ) = {(\sqrt 2 )^2} - {(\sqrt 3 )^2} $
 $
= (2 - 3) \\
= - 1 \\
$
 $ (\sqrt 2 - \sqrt 3 )(\sqrt 2 + \sqrt 3 ) = - 1. $