Question

Prove that root 5is rational

Answer 1

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

Step-by-step explanation:

Answer 2

I think there is correctìon in the question...
WE HAVE TO PROVE root 5 as irrational

So here is the solution :-
Let root 5 be rational.
then root 5 =p/q (where p and q are integers and their hcf is 1)

or
$\sqrt{5q } = q$
or
$5 {q}^{2} = {p}^{2}$

or
${q}^{2} = {p}^{2} \div 5$
since p square is divisible by 5. so p is also divisible by 5.

Similarly, prove hcf of p and f not equal to 1.

So root 5 is irrational