Prove that root 3 is an irrational number hence check whether 2-5 root 3 is rational or irrational.
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Answers
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.