Question

Prove that 2/3×root 6 is irrational numbers

Answer 1

Step-by-step explanation:

Let √6 be a rational number , then

√6 = p÷q , where p,q are integers , q not = 0 and p,q have no common factors ( except 1 )

=> 6 = p² ÷ q²

=> p² = 2q² ................(i)

As 2 divides 6q² , so 2 divides p² but 2 is a prime number

=> 2 divides p

Let p = 2m , where m is an integer .

Substituting this value of p in (i) , we get

(2m)² = 6q²

=> 2m² = 3q²

As 2 divides 2m² , 2 divides 3q²

=> 2 divides 3 or 2 divide q²

But 2 does not divide 3 , therefore , 2 divides q²

=> 2 divides q

Thus , p and q have a common factor 2 . This contradicts that p and q have no common factors ( except 1 ).

Hence , our supposition is wrong . Therefore , √6 is an irrational number.

Mark me as a brainlist