Question

prove that√3-√5 is an irrational number​

Answer 1

Prove that sqrt5 is irrational and hence prove tha-class-10 ... Let us prove that √5 is an irrational number, by using the contradiction method. So, say that √5 is a rational number can be expressed in the form of pq, where q ≠0. So, let √5 equals pq. Where p, q are co-prime integers i.e. they do not have any common factor except '1'.

Answer 2

Let  us assume on the contrary that √3- √5 is a rational number

Then , there exist coprime a and b (b≠0) ,

such That

⇒√3- √5 = a/b

⇒a/b -√3 = -√5

⇒(a/b-√3) = (-√5)  

Squaring on both Side  

⇒(a/b-√3)² = (-√5)²

⇒a²/b² - (2a√3)/b + 3 = 5

⇒a²/b² - (2a√3)/b = 2

⇒a²/b² - 2 =  (2a√3)/b

By taking LCM

⇒(a² - 2b²)/b² =  (2a√3)/b

⇒(a² - 2b²)/2ab = √3

⇒√3 is rational Number

This contradicts the fact that √3 is irrational . so our assumption is wrong

Hence √3- √5 is irrational Number