What Is The Definition of "Perfect"?

True or false :

\quad A perfect fourth power is always a perfect square.

False True

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2 solutions

Let the perfect fourth power be x 4 x^4 . We can rewrite it as ( x 2 ) 2 (x^2)^2 . So it is also a perfect square.

Ashish Menon
May 23, 2016

It is easy to understand that for all positive integers x x , x 4 = ( x 2 ) 2 x^4 = {(x^2)}^2 . So it is always a perfect square.
For x = 0 x = 0 , 0 4 = ( 0 2 ) 2 0^4 = {(0^2)}^2 so it is valid for x = 0 x = 0 too.
Even for all negative integers x x , x 4 x^4 is positive and x 2 x^2 is positive too. So, ( x 2 ) 2 (x^2)^2 is positive too and its magnitude is equal to x 4 x^4 . So, it is valid in this case too.

No need to check for decimals(non-integral) because they are not perfect \text{perfect} squares or fourth power.

So, the answer is True \color{#69047E}{\boxed{\text{True}}} .

We don't need to deal with imaginary numbers here. Perfect nth power = x^n, where x and n are real integers.

Pi Han Goh - 5 years ago

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Oh ok I just explained it. I removed it now.

Ashish Menon - 5 years ago

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