Question

show that √a+√b is an irrational number if √ab is an irrational number.

Answer 1

Step-by-step explanation:

Given:-

√ab is an irrational number.

To prove:-

show that √a+√b is an irrational number

Solution:-

let assume that √a+√b is a rational number.

=>It is in the form of p/q form (p,q are co-primes)

=>√a+√b=x/y

squaring on both sides then

=>(√a+√b)²=(x/y)²

=>a+b+2√ab=x²/y²

=>2√ab=x²/y²-(a+b)

=>√ab=[x²/y²-(a+b)]/2

=>√ab is in the form of p/q

=>√ab is a rational number

but given that √ab is an irrational number

This contradicts to our assumption

=>√a+√b is not a rational number

=>√a+√b is an irrational number

Hence,Proved.

Conclusion:-

√a+√b is an irrational number