If n = s^a*t^b, where a, b, s and t are integers, is n an integer?

Question

If  n=s^a t^b , where  a ,  b ,  s  and  t  are integers, is  \sqrt{n}  an integer?

(1)  a+b  is an even number
(2)  a  is an even number

chetan2u · Answer

The question asks whether n is square.

n=s^a t^b  
If a and b are positive even numbers, answer is yes. 
If a and b are positive odd numbers, answer will be yes when both s and t are squares.

Both statements combined tell us that an and b are even. But are they positive ??

If both are positive, say a=2 and b=4, then  n=(st^2)^2 
But if they are negative, we can have different possibilities. 
Say, a=b=-2, while s and t are 2 each, answer is no, but if s and t are 1 each, answer is yes.

E

bumpbot · Answer

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If n = s^a*t^b, where a, b, s and t are integers, is n an integer?

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