Question

prove that root 5 is irrational ​

Answer 1

let us assume to the contrary that √5 is rational

so it can be written as p/q form

√5=a/b

now we square both sides

(√5)^2=a/b^2 ( where a nd b are co prime)

5 =a^2/b^2

5b^2=a^2

b^2=a^2/5(a^2is divisible by 5 so a is also divisible by 5)

let us now assume that a =5m

√5=5m/b

now we square both the sides

(√5)^2 =5m/b^2

5=25m/b^2

5b^2=25m

b^2=25m/5

b^2=5m

b^2/5=m(b^2 id divisible by 5 so b is also divisible by 5)

This comes to conclusion that our assumption went wrong so √5 is irrational.......

(hence proved)

hope it will help you

please please mark it as brainliest

Answer 2

Step-by-step explanation:

Proof:

Let us assume that √5 is a rational number.

Sp it t can be expressed in the form p/q where p,q are co-prime integers and q≠0

⇒√5=p/q

On squaring both the sides we get,

⇒5=p²/q²

⇒5q²=p² —————–(i)

p²/5= q²

So 5 divides p

p is a multiple of 5

⇒p=5m

⇒p²=25m² ————-(ii)

From equations (i) and (ii), we get,

5q²=25m²

⇒q²=5m²

⇒q² is a multiple of 5

⇒q is a multiple of 5

Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√5 is an irrational number

Hence proved