# Prove that (a) is not arational number and (b) is not arational number.

(a) Let us assume that where a is rational.
Then, squaring both sides-

Using (a  +  b) 2 =a 2 +  2ab  +  b 2

Now, a is rational is rational and is an irrational number.
Since a rational number cannot be equal to an irrational number. Our assumption that
is rational is wrong.

(b) Let us assume that , where p and q are integers, having no common factors and q ≠ 0.

Taking cube both sides—

7q 3 = p 3 ------(i)

p is a multiple of 7
Thus p is multiple of 7.

Let p = 7m, where m is an integer.
Then, p 3 = 343 m 3 ------(ii)

7q 3 = 343 m 3 [from (i) and (ii)]
q 3 = 49 m 3 q 3 is a multiple of 7.
q is a multiple of 7.

Thus, p and q are both multiples of 7, or 7 is a factor of p and q.
This contradicts our assumption that p and q have no common factors.

Hence is not a rational 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
Application of Euclids Division Lemma50 mins
Relation Between LCM , HCF and Numbers46 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