Please Prove it?

Question

Show that (n)1/2 is irrational if n is positive integer that is not aperfect square.

Answer 1

If n^(1/2) is NOT irrational, then n^(1/2) must be rational, which means it is in the form a/b. Since n is an Integer, a and b must have some similar factor that cancels out b, leaving an integer c. Prove that n IS a perfect square here, and that n MUST be a perfect square integer to have a rational square root, leaving us with the deduction that n^(1/2) is IRRATIONAL when n is a non-perfect square integer.

Answer 2

Assume n^1/2 = a/b , where a/b is in its lowest form
Then n = a^2/b^2
If n is even, then nb^2 is even and hence a is even.
Continue showing that a/b is not in its lowest terms and hence the assumption that n^1/2 = a/b is false.Thus n^1/2 is irrational if n not perfect square.

Answer 3

suppose that sqrt(n) is rational, then we have,

sqrt(n)=a/b
n=(a^2)/(b^2)
n.b^2=a^2,

but since b^2 and a^2 are perfect squares, n should also be a perfect square to satisfy above equation.

Answer 4

Try a proof by contradiction.