When the rare coin's predicted value is approximately $82$ dollars, it is $8\%$ greater than the predicted value, in dollars, on January 1 of the previous year.

When the rare coin’s predicted value is approximately $82$ dollars, it is $8$ times the predicted value, in dollars, on January 1 of the previous year.

From the day Immanuel purchased the rare coin to $8$ years after Immanuel purchased the coin, its predicted value increased by a total of approximately $82$ dollars.

$8$ years after Immanuel purchased the rare coin, its predicted value is approximately $82$ dollars.