Question

show that root a+root b is an irrational number ìf root ab is an irrational number​

Answer 1

√a+√b is an irrational number.

Let√a+√b be a rational number. There exist two number p and q where q is not equal to 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.