Question

show that root five is irrational

Answer 1

Hey

Here is your answer,

Let root 5 be rational

Then it must in the form of p/q [q is not equal to 0][p and q are co-prime]
root 5=p/q

=> root 5 * q = p
squaring on both sides
=> 5*q*q = p*p ------> 1
p*p is divisible by 5
p is divisible by 5
p = 5c [c is a positive integer] [squaring on both sides ]
p*p = 25c*c --------- > 2
sub p*p in 1
5*q*q = 25*c*c
q*q = 5*c*c
=> q is divisble by 5

thus q and p have a common factor 5

There is a contradiction as our assumsion p &q are co prime but it has a common factor.

so √5 is an irrational.

Hope it helps you!

Answer 2

hlo friend
Let √5 is rational which is in the form of p/q where p and q have no common factor other than 1 and q is not equal to zero
then
√5=p/q
squaring both sides
5 =p^2/q^2
5q^2=p^2...............(I)

here p^2 is divisible by 5
also p is divisible by 5
let p=5m
squaring both sides we get
p^2=25m^

putting value of p^2 in equation (I) we get
5q^2=25m^2
q^2=5m^2
Again
q^2 is divisible by 5
also by fundamental theorem of arithmetic
q is divisible by 5
hence
5 is the common factor ofyp and q
so our supposition is wrong by assuming √5 as rational
therefore √5 is irrational
proved


hope it helps
if helps mark this as brainliest