Question

prove that ✓3-✓5 is irrational

Answer 1

Let root 3 + roof 5 be rational

root 3 + root 5 = P/q

(root 3 + root 5) sq=(P/q)sq

3 +5 + 2 root 15 = P sq/q Sq

root I5 = (Psq / qsq -7) 1/2

RHS is rational as all are integers
⇒ LHS is also rational but root 15 is irrational

⇒ root3 + root 5 is irrational


$i \: hope \: u \: like \: it$

Answer 2

let us assume the contrary fact that root3- root 5 is rational
root3-root5 = p/q
on squaring both side
(root3-root5 )^2 = (p/q)^2
3 +5- 2root15 =p^2/q^2
8-2root15 =p^2/q^2
2root15 =8- p^2/q^2
root 15 = 1/2 ( 8q^2 -p^2 /q^2 )
1/2 ( 8q^2 -p^2 /q^2 ) is a rational no.
so root 15 is also rational
but this contradicts the fact that root 15 is irrational
this contradiction arise due to false assumption that root 15 is a rational no.
so we conclude root 15 is irrational number

was this helpful .. pls tell