<s>[INST] What is the appropriate substitution for this equation? I need to solve this equation:
$x^\frac{1}{3}+x^\frac{1}{7}=2(x^5)^\frac{1}{21}.$
I can't find the substitution which reduces it to a system or equation without roots.
 [/INST] $$\text{The equation can be rewritten as follows:}$$
$$x^{\frac{7}{21}} + x^{\frac{3}{21}} = 2(x^{\frac{5}{21}})$$
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$$\text{Now, let } y = x^{\frac{1}{21}}:$$
$$y^7 + y^3 = 2y^5 \implies y^7 - 2y^5 + y^3 = 0$$
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$$\text{If } y = 0, \text{ then } x = 0, \text{ else we have: }$$
$$y^4 - 2y^2 + 1 = 0 \implies (y^2 - 1)^2 = 0 \implies y^2 = 1$$
$$$$
$$\text{Hence, } y \in \{-1, 0, 1\}.$$
$$\text{And thus, } x \in \{-1, 0, 1\} \text{ since } x = y^{21}.$$
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