Question

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Q. Prove that
${p}^{ \frac{1}{n} }$
is irrational when p is prime and
n > 1.

Answer 1

As p is prime, p has only two factors.

And we know from the question that n >1.

Let us suppose that the result thus obtained would be rational

Therefore,

${p}^{\frac{1}{n} }$ = q.

For some rational number q.

Therefore,

p = ${q}^{n}$

Therefore there are is a rational number q that is a factor of p.

But this contradicts the fact that p is a prime number.

Therefore, we can say that no such rational number exists and the value of the result thus would be an irrational number.

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Answer 2

Here your answer:-

'p' is prime number then it has no cube root or square root or any root value.
if N>1 then p will be written as
' n root p' we put p in root of n but p can not be taken out from root because it is a prime number.
So, it is irrational number.
Hence,prooved.



Example related to ques.:-

Let a prime number 7 will root value 3 which is greater than 1.

there is no cube root of 7 so it is also irrational number.





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