Question

prove that root3-root2 be a irrational no

Answer 1

as root 3 and root 2 irrational therefore there subtraction is also irrational

Answer 2

Let √3 and √2 be rational numbers
∴√3=p/q ; q>0 ; (p,q) = 0
=> √3q = p
( √3q )² = p² [ squaring both sides ]
3q² = p² .............. ..- (1)
⇒p² is divisible by 3
⇒p is divisible by 3
p=3m
p²= (3m)²
3q²= 9m² [ From 1 ]
q² = 3m²
⇒q² is divisible by 3
⇒q is divisible by 3
∴ (p,q) = 3
—————-> <————————-
∴ √3 is an irrational number. - (2)
Similarly √2 is an irrational number. - (3)

(2) - (3)
√3 - √2 is an irrational number ( ∵ irrational no. - irrational no. = irrational no. )