Question

prove that root 3 is irrational. also prove that 7-2 root 3 is an irrational

Answer 1

Let sqrt3 be a rational number. Let p and q be coprime integers.

Let p/q = sqrt(3) = reduced form of rational number. .
p * p = 3 * q *q.
So on the LHS, p must have a factor 3. Let p=3n.

So 3 n*n = q*q
Now on the RHS q must have a factor 3. So q=3m.

n*n = 3*m*m.
So it turns out that p and q are not coprime.
It's proved by contradiction that sqrt3 is irrational .
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See the enclosed picture for the other part.

Answer 2

Answer:

Hi

Step-by-step explanation:

Let us assume that √3+√7 is rational.

That is , we can find coprimes a and b (b≠0) such that

Therefore,

Squaring on both sides ,we get

Rearranging the terms ,

Since, a and b are integers , is rational ,and so √3 also rational.

But this contradicts the fact that √3 is irrational.

This contradiction has arisen because of our incorrect assumption that √3+√7 is rational.

Hence, √3+√7 is irrational.

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