Juan purchased an antique that had a value of \$200$200dollar sign, 200 at the time of purchase. Each year, the value of the antique is estimated to increase 10\, percent over its value the previous year. The estimated value of the antique, in dollars, 222 years after purchase can be represented by the expression 200a200a200, a, where aaa is a constant. What is the value of aaa?