x^6= 1/(x^3)

Solution

x^{6} = \frac{1}{x ^{3}}

x = 1, x = - \frac{1}{2} + \frac{3}{2} i, x = - \frac{1}{2} - \frac{3}{2} i, x = cos (\frac{2 π}{9}) + i sin (\frac{2 π}{9}), x = cos (\frac{8 π}{9}) + i sin (\frac{8 π}{9}), x = cos (\frac{14 π}{9}) + i sin (\frac{14 π}{9}), x = cos (\frac{2 π}{9}) - i sin (\frac{2 π}{9}), x = cos (\frac{4 π}{9}) + i sin (\frac{4 π}{9}), x = cos (\frac{10 π}{9}) + i sin (\frac{10 π}{9})

Solution steps

x^{6} = \frac{1}{x ^{3}}

Multiply both sides by

x^{3}

x^{9} = 1

Solve

x^{9} = 1 : x = 1, x = - \frac{1}{2} + \frac{3}{2} i, x = - \frac{1}{2} - \frac{3}{2} i, x = cos (\frac{2 π}{9}) + i sin (\frac{2 π}{9}), x = cos (\frac{8 π}{9}) + i sin (\frac{8 π}{9}), x = cos (\frac{14 π}{9}) + i sin (\frac{14 π}{9}), x = cos (\frac{2 π}{9}) - i sin (\frac{2 π}{9}), x = cos (\frac{4 π}{9}) + i sin (\frac{4 π}{9}), x = cos (\frac{10 π}{9}) + i sin (\frac{10 π}{9})

x = 1, x = - \frac{1}{2} + \frac{3}{2} i, x = - \frac{1}{2} - \frac{3}{2} i, x = cos (\frac{2 π}{9}) + i sin (\frac{2 π}{9}), x = cos (\frac{8 π}{9}) + i sin (\frac{8 π}{9}), x = cos (\frac{14 π}{9}) + i sin (\frac{14 π}{9}), x = cos (\frac{2 π}{9}) - i sin (\frac{2 π}{9}), x = cos (\frac{4 π}{9}) + i sin (\frac{4 π}{9}), x = cos (\frac{10 π}{9}) + i sin (\frac{10 π}{9})

Verify Solutions

Find undefined (singularity) points:

x = 0

Combine undefined points with solutions:

x = 1, x = - \frac{1}{2} + \frac{3}{2} i, x = - \frac{1}{2} - \frac{3}{2} i, x = cos (\frac{2 π}{9}) + i sin (\frac{2 π}{9}), x = cos (\frac{8 π}{9}) + i sin (\frac{8 π}{9}), x = cos (\frac{14 π}{9}) + i sin (\frac{14 π}{9}), x = cos (\frac{2 π}{9}) - i sin (\frac{2 π}{9}), x = cos (\frac{4 π}{9}) + i sin (\frac{4 π}{9}), x = cos (\frac{10 π}{9}) + i sin (\frac{10 π}{9})

\frac{4}{0.7071 x} = 3

\frac{3 x}{x - 2} = \frac{6}{x - 2} + 4

\frac{2 π}{3} = \frac{2 π}{b}

x^{- 1} = - \frac{1}{4}

\frac{8}{5 a} = - 6