Notice that Graph A is a circle. We can see that this is a true representation of the equation by checking some of the points: x2=1y2
By rewriting the equation in this way, we see that when x=0, y=1 or y=1 and when x=1 or x=1, y=0. But what happens if you choose a value for x which is larger than 1? Try the value x=2. We know that x2=22=(2)(2)=4, but how can 1y2=4? This could only be possible if y2=3. However, since a number raised to an even power (for example 2, 4, 6,...) can never be negative, we know that our Graph A is restricted to the values 1≤x≤1 and 1≤y≤1.
Notice that Graph B, although it is a similar equation, does not create a closed shape. This graph is closer to a line with a rounded portion. Before examining Graph B in depth, let’s compare the two equations and see what their similarities and differences are. Recall the first equation, x2+y2=1, and now look at the second equation, x3+y3=1.
What do the two equations have in common?

‣Both equations are equal to 1

‣Both equations are the sum of two variables which have been raised to the same power
How are the two equations different?

‣One equation raises its variables to an even power and the other equation raises its variables to an odd power
We can see now that the most outstanding difference between these two equations is the value of the exponent. Let’s examine what happens when a number is cubed (or raised to an odd power such as 3, 5, 7,...). Let x=2, then x3=23=(2)(2)(2)=8. Now let x=2. Recall that if a negative number is squared the result will always be a positive number: (2)2=(2)(2)=4. However, (2)3=(2)(2)(2)=8. This means that Graph B will not be restricted like Graph A was. Let’s check some of the points of Graph B: x3=1y3. We immediately can see a difference from the first equation: When x=0, y=1 and when x=1, y=0, however when x=1, y3=2 and when y=1, x3=2. Unlike in the first equation, it is possible to find a number whose cube is 2 or 2. Therefore Graph B is not restricted like Graph A.
Now let’s examine two more equations: x4+y4=1 and x5+y5=1; we will call them Graphs C and D, respectively.