Find roots of:
$$x^{6}\ -\ \left(x-1\right)^{6}=0 \tag {1}$$
I know this equation has $4$ complex roots and exactly one real roots of value $0.5$.
However, my first instinct was to do this:
$$x^{6}\ =\ \left(x-1\right)^{6} \tag{2}$$
"raise both sides to 6-th power" to get:
$$x=x-1\tag{3}$$
Which has no real solution. I see that this wrong. How to avoid this error? Thanks.
Inspired by watching this youtube video
Edit:
I am not asking about how to solve the problem. I want to know
what I did wrong from an Algebraic stand-point. Maybe raising to the power? What is wrong with that?
Best Answer
Thanks for all the posted comments above. At the moment, no one had posted an answer, but I understood the following, which combined may provide an answer.
$$|x|=|x-1|$$
can't always be written as $x=x-1$. I need to learn how to solve such an equation.
Also, $$a^n=b^n$$
Does not always imply that $a=b$. The result is affected by the domain of $a$ and $b$ and whether the power is even or odd or integer or not, maybe among other factors.