<s>[INST] Question about equivalent equations I am solving equations with radicals using this substitution:
$f(x)^{\frac{1}{3}}+g(x)^{\frac{1}{3}}=h(x)$
$f(x)+g(x)+3(f(x)g(x))^{\frac{1}{3}}(f(x)^{\frac{1}{3}}+g(x)^{\frac{1}{3}})=h(x)^{{3}}$
Now I substitute the expression $f(x)^{\frac{1}{3}}+g(x)^{\frac{1}{3}}$  with $h(x)$. Why after this substitution the new equation is not equivalent to the original??
 [/INST] To make things clear, let's denote $a = f(x), b = g(x), c = h(x)$, then
$$a^{1/3}+b^{1/3} = c$$
$$(a^{1/3}+b^{1/3})^3 = a + 3a^{2/3}b^{1/3} + 3a^{1/3}b^{2/3} + b = a+3a^{1/3}b^{1/3}(a^{1/3} + b^{1/3}) + b =$$
$$= a+3a^{1/3}b^{1/3}c + b = a+3(ab)^{1/3}c + b =c^3.$$
Therefore
$$(a+3(ab)^{1/3}c + b)^{1/3} = c = a^{1/3} + b^{1/3}$$
or
$$ a+ b =c^3-3(ab)^{1/3}c = c(c^2-3(ab)^{1/3}).$$
Because you are mixing the terms, you can get some messy equations, but it does not mean they are wrong.
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