Question

if root ab is an irrational number prove that root a plus root b is irrational number​

Answer 1

Answer:

This is the contradiction to our assumption. Hence a+√b is an irrational number.

Step-by-step explanation:

Let a+√b be a rational number. There exist two number p and q where q ≠0 and p,q are co prime i.e. p/q =a+√b

Then,

(p/q)² = (a+√b)² [squaring both sides]

=› p²/q =a²+b/q ————(¹)

Since p and q are co prime L.H.S. is always fractional and R.H.S. is always integer . If q =1,the equation (¹) is hold good but it was impossible that there was no number whose square is a² +b .

Let a+√b be a rational number. There exist two number p and q where q ≠0 and p,q are co prime i.e. p/q =a+√b

Then,

(p/q)² = (a+√b)² [squaring both sides]

=› p²/q =a²+b/q ————(¹)

Since p and q are co prime L.H.S. is always fractional and R.H.S. is always integer . If q =1,the equation (¹) is hold good but it was impossible that there was no number whose square is a² +b .

This is the contradiction to our assumption. Hence a+√b is an irrational number.