What are the solutions to the equation f=f^9 ?

The solutions to the equation f=f^9 are

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f=f^9

f = f^{9}

f = 0, f = - \frac{2}{2} + i \frac{2}{2}, f = - \frac{2}{2} - i \frac{2}{2}, f = \frac{2}{2} + i \frac{2}{2}, f = \frac{2}{2} - i \frac{2}{2}, f = i, f = - i, f = - 1, f = 1

Solution steps

f = f^{9}

Switch sides

f^{9} = f

Move

f

to the left side

f^{9} - f = 0

Factor

f^{9} - f : f (f^{2} + 2 f + 1) (f^{2} - 2 f + 1) (f^{2} + 1) (f + 1) (f - 1)

f (f^{2} + 2 f + 1) (f^{2} - 2 f + 1) (f^{2} + 1) (f + 1) (f - 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

f = 0 or f^{2} + 2 f + 1 = 0 or f^{2} - 2 f + 1 = 0 or f^{2} + 1 = 0 or f + 1 = 0 or f - 1 = 0

Solve

f^{2} + 2 f + 1 = 0 : f = - \frac{2}{2} + i \frac{2}{2}, f = - \frac{2}{2} - i \frac{2}{2}

Solve

f^{2} - 2 f + 1 = 0 : f = \frac{2}{2} + i \frac{2}{2}, f = \frac{2}{2} - i \frac{2}{2}

Solve

f^{2} + 1 = 0 : f = i, f = - i

Solve

f + 1 = 0 : f = - 1

Solve

f - 1 = 0 : f = 1

The solutions are

f = 0, f = - \frac{2}{2} + i \frac{2}{2}, f = - \frac{2}{2} - i \frac{2}{2}, f = \frac{2}{2} + i \frac{2}{2}, f = \frac{2}{2} - i \frac{2}{2}, f = i, f = - i, f = - 1, f = 1

f (x) = x^{7}

f (x) = x^{2} - 4 x - 4

f (x) = x^{\frac{2}{5}}

f (x) = 3 x - 8

f (x) = 3 x + 8

What are the solutions to the equation f=f^9 ?
The solutions to the equation f=f^9 are
Find the zeros of f=f^9
The zeros of f=f^9 are