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Question Number 132285 by bemath last updated on 13/Feb/21

Simplify the equation of

\frac{(x^{\frac{1}{3}} - x^{\frac{1}{6}}) (x^{\frac{1}{2}} + x) (x^{\frac{1}{2}} + x^{\frac{1}{3}} + x^{\frac{2}{3}})}{(x^{\frac{4}{3}} - x) (x + x^{\frac{1}{3}} + x^{\frac{2}{3}})}

with x \neq 0

Answered by Olaf last updated on 13/Feb/21

X = \frac{(x^{\frac{1}{3}} - x^{\frac{1}{6}}) (x^{\frac{1}{2}} + x) (x^{\frac{1}{2}} + x^{\frac{1}{3}} + x^{\frac{2}{3}})}{(x^{\frac{4}{3}} - x) (x + x^{\frac{1}{3}} + x^{\frac{2}{3}})}

Let a = x^{\frac{1}{6}}

X = \frac{(a^{2} - a) (a^{3} + a^{6}) (a^{3} + a^{2} + a^{4})}{(a^{8} - a) (a^{6} + a^{2} + a^{4})}

X = \frac{a (a - 1) a^{3} (a^{3} + 1) a^{2} (a^{2} + a + 1)}{a (a^{7} - 1) a^{2} (a^{4} + a^{2} + 1)}

X = \frac{a^{3} (a - 1) (a^{3} + 1) (a^{2} + a + 1)}{(a^{7} - 1) (a^{4} + a^{2} + 1)}

X = \frac{a^{3} (a - 1) (a + 1) (a^{2} - a + 1) (a^{2} + a + 1)}{(a^{7} - 1) (a^{4} + a^{2} + 1)}

X = \frac{a^{3} (a - 1) (a + 1) (a^{4} + a^{2} + 1)}{(a^{7} - 1) (a^{4} + a^{2} + 1)}

X = \frac{a^{3} (a - 1) (a + 1)}{a^{7} - 1} = \frac{a^{3} (a^{2} - 1)}{a^{7} - 1}

X = \frac{x^{\frac{1}{2}} (x^{\frac{1}{3}} - 1)}{x^{\frac{7}{6}} - 1}

\dots or X = \frac{a^{3} (a + 1)}{a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a + 1}

Answered by benjo_mathlover last updated on 13/Feb/21

\frac{x^{\frac{1}{6}} (x^{\frac{1}{6}} - 1) x^{\frac{1}{2}} (1 + x^{\frac{1}{2}}) (x^{\frac{1}{2}} + x^{\frac{1}{3}} + x^{\frac{2}{3}})}{x (x^{\frac{1}{3}} - 1) (x + x^{\frac{1}{3}} + x^{\frac{2}{3}})}

= \frac{(x^{\frac{1}{6}} - 1) (1 + x^{\frac{1}{2}}) (x^{\frac{1}{2}} + x^{\frac{1}{3}} + x^{\frac{2}{3}})}{x^{\frac{1}{3}} (x^{\frac{1}{6}} - 1) (x^{\frac{1}{6}} + 1) (x + x^{\frac{1}{3}} + x^{\frac{2}{3}})}

= \frac{(1 + x^{\frac{1}{2}}) (x^{\frac{1}{2}} + x^{\frac{1}{3}} + x^{\frac{2}{3}})}{x^{\frac{1}{3}} (x^{\frac{1}{6}} + 1) (x + x^{\frac{1}{3}} + x^{\frac{2}{3}})}

= \frac{(1 + x^{\frac{1}{2}}) (x^{\frac{1}{6}} + 1 + x^{\frac{1}{3}})}{(x^{\frac{1}{6}} + 1) (x + x^{\frac{1}{3}} + x^{\frac{2}{3}})}