3. Solve the equation

$$
\frac{x^{7}-1}{x^{5}-1}=\frac{x^{5}-1}{x^{3}-1}
$$
Answer: $0, -1$.

Solution. For $x \neq 1$, both denominators are non-zero, and by the rule of proportion, the given equation takes the form

$$
\left(x^{7}-1\right)\left(x^{3}-1\right)=\left(x^{5}-1\right)^{2} \Leftrightarrow x^{7}-2 x^{5}+x^{3}=0 \Leftrightarrow x^{3}\left(x^{2}-1\right)^{2}=0 \Leftrightarrow x^{3}(x-1)^{2}(x+1)^{2}=0
$$

The left side of the last equation is zero when $x=-1, 0$, and 1, but only two values are included in the answer: $x=-1$ and 0, since at $x=1$ the original fractions are undefined.