Question

Prove that 2√2 is irrational after proving √2 is irrationl​

Answer 1

So to prove √2 is irrational, we use the contradiction method

Let's assume that √2 is not irrational... That is, it is rational and can be written in the form of p/q.

so, √2=p/q

so, √2q=p

now, squaring on both sides, we get:

2q^2=p^2 - let this be equation (i)

so, 2 is a factor of p^2 Therefore, it is also a factor of p.

Now, 2*(something) =p (since 2 is a factor of p)

let something=m

so, 2m= p

Now, substituting p value in (i)

2q^2= (2m)^2

2q^2=4m^2

so, q^2=2m^2

This means 2 is also a factor of q.

But in a rational no. p/q, p and q are co-primes.

This contradicts the fact that 2 is factor of both p and q and hence, √2 is irrational.

Now, we have to prove that 2√2 is irrational.

Like we did previously, assume 2√2 is not irrational hence,

2√2=p/q

so, √2= p/2q

This shows us that √2 is rational but we have already proved that it is irrational. So, our assumption is wrong and hence, 2√2 isbirrational.

THATS IT!!! I HOPE I WAS ABLE TO HELP U... IF YES PLZ MARK AS BRAINLIEST..