This result bears much similarity to my treatment of the Irrationality of Fermat’s Last Theorem, however is much more powerful due to its greater range, efficiency and applicability. It also is a method for the proof of irrationality. Although it is a simple argument it is quite far reaching. I’ll make you wait in anticipation no longer, only working in R:
x ∈ Q
x ∉ J
x = m/p (where m and p share no prime factors)
xn = mn/pn
Since the prime factors of mn will be the same as those for m (because of unique factorization mn can be made up by multiplying the prime factors of m by themselves – e.g 12 = 2x2x3, 144 = 2x2x2x2x3x3), and the same can be said for pn and p,
xn ∉ J
∴ The nth root of a number cannot be a rational fraction, since the nth power of a fraction is always another fraction. So, if an integer is not the nth power of another integer, its nth root is irrational.
x ∉ J
x = m/p (where m and p share no prime factors)
xn = mn/pn
Since the prime factors of mn will be the same as those for m (because of unique factorization mn can be made up by multiplying the prime factors of m by themselves – e.g 12 = 2x2x3, 144 = 2x2x2x2x3x3), and the same can be said for pn and p,
xn ∉ J
∴ The nth root of a number cannot be a rational fraction, since the nth power of a fraction is always another fraction. So, if an integer is not the nth power of another integer, its nth root is irrational.
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