If you first add to both sides of the equation and re-write the fractional exponents as radicals:
You can readily see that this equation has no solution. That's because over the most restrictive domain (the left hand side function), namely
,
The values of the two sides of the equation will differ by a factor of at the low end of the domain and will differ by a decreasing amount as x gets larger, but the difference can never be exactly zero.
This is the reason that using ordinary solution methods on this equation results in an absurdity:
Square both sides:
Add -x to both sides:
Absurd.
Therefore, the solution set of the given equation is the empty set.
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