[ Examples and counterexamples. Constructions ]

It is known about numbers $a$ and $b$ that $a=b+1$. Can it happen that $a^{4}=b^{4}$?
Let $a=1 / 2, b=-1 / 2$, then $a^{4}=b^{4}=\frac{1}{16}$. It can be proven that this example is unique (this is not required from students). Indeed, $a^{4}=b^{4} \Leftrightarrow|a|=|b|$. The case $a=b$ is impossible, the case $a=-b$ gives the specified example.

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