Question

there is a no. x such that x square is irrational but x power 4 is rational then x can b​

Answer 1

Answer:

Yes, in general there is at least one such number. More generally there should be an infinite number of such numbers, though none of them would be integers.

In general, if we take the fourth root of any prime number,

X=P−−√4 ,

we should find that,

X2=P−−√

is also irrational. Why? Because if X could be expressed as a rational number, say the ratio of integers m over n , then we could write,

P=X4=m4n4.

But we have already determined that P cannot have any integer factors other than itself and 1. Yet all prime numbers are greater than 1, so each factor of P , given as m/n , would need to be first greater than 1. Second, if we multiply both sides of the above equation by n4, we have,

m4=P×n4 .

But whatever number n is, it could only have an integer number of factors of P inside it, say Pk . And the same for m , say Pj . Now, looking only at the number of factors of P on the left and the right, we have,

P4j=P4k+1 .

And whatever integers j and k are, being integers they cannot produce an equality between the left and right sides because of the extra factor of P on the right side.

So any prime number will also be a solution to this problem. And any products of primes will also be solutions, as long as they are not to the fourth powers of P.