Question

please answer my question​

Answer 1

Answer:

Let us consider

3

be a rational number, then

3

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



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

So,

3=p

2

/q

2

p

2

=3q

2

…. (1)

As we know, ‘3’ divides 3q

2

, so ‘3’ divides p

2

as well. Hence, ‘3’ is prime.

So 3 divides p

Now, let p=3k, where ‘k’ is an integer

Square on both sides, we get

p

2

=9k

2

3q

2

=9k

2

[Since, p

2

=3q

2

, from equation (1)]

q

2

=3k

2

As we know, ‘3’ divides 3k

2

, so ‘3’ divides q

2

as well. But ‘3’ is prime.

So 3 divides q

Thus, p and q have a common factor of 3. This statement contradicts that ‘p’ and ‘q’ has no common factors (except 1).

We can say that

3

is not a rational number.

3

is an irrational number.

Now, let us assume (2/5)

3

be a rational number, ‘r’

So, (2/5)

3

=r

5r/2=

3

We know that, ‘r’ is rational, ‘5r/2’ is rational, so ‘

3

’ is also rational.

This contradicts the statement that

3

is irrational.

So, (2/5)

3

is an irrational number.

Hence proved.