7b98c4…: “Up to the free cube”

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Pick two numbers positve real numbers $a$ and $b$ so that $a \approx b$ . Pick a very large integer $N$ . What can you say about $a^N$ and $b^N$ ?

$a^N$ and $b^N$ must both be large because $N$ is large. $a^N$ and $b^N$ must both be small because $|a-b|^N$ is small. $a^N$ must be very close to $b^N$ because $a \approx b$ . $a^N$ might not be very close to $b^N$ even though $a \approx b$ .