**Assignment #1**

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**Problem:**

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After graphing:

x2 + y2 = 1

x3 + y3 = 1

x4 + y4 = 1

x5 + y5 = 1

What is expected
for the graphs of:

x24 + y24 = 1

or

x25 + y25 = 1?

First, letŐs
analyze the original graphs:

x^{3} + y^{3
}= 1

x^{4} + y^{4
}= 1

x^{5} + y^{5
}= 1

The graphs begin
to show that as the exponents of x and y increase the curves centered about the
origin begin to develop into right angles. The original circle begins to
develop into a square as the exponents increase in even numbers. See the
graph below of increasing even exponents:

x^{2} + y^{2 }= 1

x^{4} + y^{4 }= 1

As the exponents
increase in odd numbers, the original circular break in the line y = -x becomes
more square. See the graph below of increasing odd exponents:

x^{3} + y^{3 }= 1

x^{5} + y^{5 }= 1

x^{7} + y^{7 }= 1

According to
these observations, the graph of

x^{24} + y^{24} = 1

should resemble
a square, and the graph of

x^{25} + y^{25} = 1

should have a
break resembling half of a square in the line y = -x. Below is our
predicted graph:

x^{24}+y^{24 }= 1

x^{25}+y^{25 }= 1