# If p and q are co-prime positive integers then prove that is an irrational number.

Let √p + √q be a rational number.

Let √p + √q = x where x is integral number.

Squaring both sides.

(√p + √q)2 = x2

p + q + 2√pq = x2

Since p and q are co-prime positive integers. So root of p and root of q will definitely be an irrational numbers as they are not perfect squares. So √pq has to be an irrational number.

So the assumption made at the beginning of the problem is false.

So it is proved that √p + √q is an irrational number.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Interactive Quiz:Euclid's Division Lemma44 mins
Fundamental Theorem of Arithmetic-238 mins
NCERT | Imp. Qs. on Rational and Irrational Numbers44 mins
Fundamental Theorem of Arithmetic- 143 mins
Champ Quiz | Fundamental Principle Of Arithmetic41 mins
Euclids Division Lemma49 mins
Relation Between LCM , HCF and Numbers46 mins
Application of Euclids Division Lemma50 mins
Quiz | Fun with Fundamental Theorem of Arithmetic51 mins
Quiz | Imp Qs on Real Numbers37 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses