x^{-8}=1

Solution

x^{- 8} = 1

x = 1, x = - 1, x = i, x = - i, x = \frac{1}{2} + \frac{1}{2} i, x = - \frac{1}{2} - \frac{1}{2} i, x = \frac{1}{2} - \frac{1}{2} i, x = - \frac{1}{2} + \frac{1}{2} i

Solution steps

x^{- 8} = 1

Refine

\frac{1}{x ^{8}} = 1

Multiply both sides by

x^{8}

1 = x^{8}

Solve

1 = x^{8} : x = 1, x = - 1, x = i, x = - i, x = \frac{1}{2} + \frac{1}{2} i, x = - \frac{1}{2} - \frac{1}{2} i, x = \frac{1}{2} - \frac{1}{2} i, x = - \frac{1}{2} + \frac{1}{2} i

x = 1, x = - 1, x = i, x = - i, x = \frac{1}{2} + \frac{1}{2} i, x = - \frac{1}{2} - \frac{1}{2} i, x = \frac{1}{2} - \frac{1}{2} i, x = - \frac{1}{2} + \frac{1}{2} i

Verify Solutions

Find undefined (singularity) points:

x = 0

Combine undefined points with solutions:

x = 1, x = - 1, x = i, x = - i, x = \frac{1}{2} + \frac{1}{2} i, x = - \frac{1}{2} - \frac{1}{2} i, x = \frac{1}{2} - \frac{1}{2} i, x = - \frac{1}{2} + \frac{1}{2} i

0 = \frac{x ^{3} - 8}{2 x ^{2}}

0 = \frac{3 x}{x - 2}

\frac{13}{60} = \frac{91}{x}

x + \frac{1}{x} = \frac{a}{60}

12 = x + \frac{36}{x}