If k is even number then for what value of x and y, x2k−(xy)k+y2k=0

Cassarrim1

Cassarrim1

Answered

2022-02-01

If k is even number then for what value of x and y,
x2k(xy)k+y2k=0

Answer & Explanation

nebajcioz

nebajcioz

Expert

2022-02-02Added 15 answers

Suppose y is zero, then the only solution to x2k=0 is x=0. Thus, x=y=0 is one of the possible solutions.
Now assume y is not zero. Then dividing everything by y2k you get
(xy)2k(xy)k+1=0
or, after substituting z=(xy)k,
z2z+1=0
This quadratic equation has no real solutions.
This means x=y=0 is the only possible solution, and even that for k>0 because when k=0 the expression 00 is not well-defined.

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