$x^n + y^n =1$
for each $n \in \{1, 2, 1000\}.$

Try to consider the case when both
$x$
and$y$
are positive, for$n=1.$

Can the behaviour in the first quadrant be reproduced for the other three quadrants, given we take absolute values of both
$x$
and$y?$

What is the formula for the distance from a given point to the origin? Does it look familiar?

How small is
$0.1^{1000}?$
What about$0.5^{1000}$
or even$0.99^{1000}?$

The equation
$x^n+y^n=1$
will yield a symmetric shape with respect to both$x$
and$y$
axes (why?). Thus we can focus only on the first quadrant where$x$
and$y$
are both positive i.e. where$x^n+y^n=1,$
and then reflect the curve onto the other three quadrants.For
$n=1,$
we have$y=1x.$
Reflecting this onto other quadrants forms a diamond with corners at$(\pm1,0)$
and$(0,\pm1).$
For
$n=2,$
we obtain an equation of the unit circle. This is true, because$x^2+y^2$
is the formula for the squared distance from a given point to the origin.For
$n=1000,$
think about what happens to any number strictly less than 1 when raised to a very big power. It tends to 0. Now consider all values of$x$
that are sufficiently less than 1, even up to 0.999; these will cause$x^n$
to decrease sharply towards$0,$
and$y^n$
to raise sharply towards$1,$
for all such$x.$
Thus the graph tends to become a square.