Products of Perfect Powers

Number Theory Level 2

True or false:

The product of any two perfect n -th powers will always produce another perfect n -th power.

False True

Domenic Vlachos by Domenic Vlachos , 18, Australia

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2 solutions

X X
Apr 26, 2018

a n b n = ( a b ) n a^nb^n=(ab)^{n}

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Theodore Sinclair
Apr 26, 2018

The definition of an nth power is that each power of a prime factor in its prime factorisation is a multiple of n. When we multiply together two perfect nth powers it is like adding multiples of n together (including 0). A multiple of n + another multiple of n will always = a multiple of n. We can prove this easily: xn+yn=n(x+y).

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