Question

prove that root 3 + root 5 is irrational no.

Answer 1

To prove: √3+√5 is irrational

To prove it let us assume it to be a rational number

Rational numbers are the ones which can be expressed in p/q form where p,q are integers and q isn't equal to 0.

√3+√5=p/q

√3=(p/q)-√5

Squatting on both sides.

3=p²/q²-(2√5p)/q+5

(2√5p)/q=5-3-p²/q²

2√5p/q=(2q²-p²)/q²

√5=(2q²-p²)/q²*q/2p

√5=(2q²-p²)/2pq

As p and q are integers RHS is rational

As RHS is rational LHS is also rational i.e √5 is rational

But this contradicts the fact that √5 is irrational.

This contradiction arosebecause of our false assumptionSo

√3+√5 is irrational.

Answer 2

hey!

here is the answer

this regards by utsav

√3+√5 is irrational

prove that ....


solution :

let √3+√5 be a rational number, then 3+√5 = a/b where a and b are integers and b is not equal to zero..

squaring both side , we get

( √3+√5)^2 =(a/b)^2

°-° let's see what happens

here we use the formula (a+b)^2

then (√3)^2 +2×√3×√5+(√5)^2 =a^2/b^2

3+2√15+5=a^2/b^2

8+2√15 = a^2/b^2

2√15= a^2/b^2 -8

2√15= a^2-8b^2/b^2

√15= a^2-8b^2 /2b^2
here a^2-8b^2/2b^2 is a rational number

but √15 is irrational number

so,the contradiction we supposed is wrong

hence , √3+√5 is an irrational number

hope it helps

thanx