prove that 3+2√2 is irrational number .given √2 is a irrational no.​

Step-by-step explanation:

Let us consider

2

be a rational number, then

2

=p/q, where ‘p’ and ‘q’ are integers, q



=0 and p, q have no common factors (except 1).

So,

2=p

2

/q

2

p

2

=2q

2

…. (1)

As we know, ‘2’ divides 2q

2

, so ‘2’ divides p

2

as well. Hence, ‘2’ is prime.

So 2 divides p

Now, let p=2k, where ‘k’ is an integer

Square on both sides, we get

p

2

=4k

2

2q

2

=4k

2

[Since, p

2

=2q

2

, from equation (1)]

q

2

=2k

2

As we know, ‘2’ divides 2k

2

, so ‘2’ divides q

2

as well. But ‘2’ is prime.

So 2 divides q

Thus, p and q have a common factor of 2. This statement contradicts that ‘p’ and ‘q’ has no common factors (except 1).

We can say that

2

is not a rational number.

2

is an irrational number.

Now, let us assume 3−

2

be a rational number, ‘r’

So, 3−

2

=r

3–r=

2

We know that, ‘r’ is rational, ‘3−r’ is rational, so ‘

2

’ is also rational.

This contradicts the statement that

2

is irrational.

So, 3−

2

is an irrational number.

Hence proved.

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