True or false :
A perfect fourth power is always a perfect square.
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It is easy to understand that for all positive integers
x
,
x
4
=
(
x
2
)
2
. So it is always a perfect square.
For
x
=
0
,
0
4
=
(
0
2
)
2
so it is valid for
x
=
0
too.
Even for all negative integers
x
,
x
4
is positive and
x
2
is positive too. So,
(
x
2
)
2
is positive too and its magnitude is equal to
x
4
. So, it is valid in this case too.
No need to check for decimals(non-integral) because they are not perfect squares or fourth power.
So, the answer is True .
We don't need to deal with imaginary numbers here. Perfect nth power = x^n, where x and n are real integers.
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Let the perfect fourth power be x 4 . We can rewrite it as ( x 2 ) 2 . So it is also a perfect square.