A rise in the nominal exchange rate ($/€) represents a depreciation of the dollar relative to the euro, but a rise in the real exchange rate ($/€) represent an appreciation of the dollar. Explain why this is true

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The nominal exchange rate represents the number of dollars required to buy one euro. An increase in that number means that more dollars are required to purchase a euro. The dollar price of the euro has risen, and the dollar has depreciated because one dollar now buys fewer euros. However, holding nominal rates constant, the reverse is true about real exchange rates. The real exchange rate is the nominal rate times the ratio of the foreign price level to the domestic price level. For the real rate to increase, the foreign price level must rise more slowly than the domestic price level. All other things equal, that means that a dollar can buy more goods in Europe than it could in the U.S. The dollar has thus appreciated in real terms, because it can purchase more goods abroad than before.