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Prove that $ 2\sqrt 5 - \sqrt 7 $ is irrational.

Last updated date: 21st Jun 2024
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Hint: To prove the given expression as an irrational, we will use the contradiction method in which we will assume something and then solve it. We will just try to prove that whatever we assumed was wrong. Hence, if the assumed situation is wrong then ultimately the actual thing will be converse of it.

Complete step-by-step answer:
A rational number is the number which can be expressed as the ratio of two numbers or which can be expressed as the p/q form or as the quotient or the fraction with non-zero denominator whereas, the numbers which are not represented as the rational are known as the irrational number.
Let us assume that the given expression $ 2\sqrt 5 - \sqrt 7 $ is rational.
This means that- the given number can be expressed in the form of $ \dfrac{p}{q} $ where p and q belong to the integers and it can be composite or prime numbers.
So, $ 2\sqrt 5 - \sqrt 7 = \dfrac{p}{q} $
Move the term from one side to another. When you move any term from one side to another, the sign also changes. Negative terms become positive and vice-versa.
 $ \Rightarrow 2\sqrt 5 = \dfrac{p}{q} + \sqrt 7 $
Simplify the above equation –
 $ \Rightarrow 2\sqrt 5 = \dfrac{{p + q\sqrt 7 }}{q} $
When any term multiplicative on one side is moved to the opposite side, then it goes to the denominator.
 $ \Rightarrow \sqrt 5 = \dfrac{{p + q\sqrt 7 }}{{2q}} $
Since, $ \sqrt 5 $ is an irrational number and therefore it is not possible for any irrational number equal to rational number.
Irrational number $ \ne $ Rational number
Therefore, if the left hand side of the equation is irrational then the right hand side of the equation is also irrational.
Hence, our assumption that the given number is rational is not true.
Hence, proved that $ 2\sqrt 5 - \sqrt 7 $ is irrational.

Note: Know the difference between the rational and irrational numbers properly. The set of the real numbers is made by combining the set of all the rational numbers and all the set of irrational numbers. The real numbers include natural numbers, whole numbers, integers, and the rational numbers including all the fractions, decimals and repeating numbers and irrational numbers.