Unfortunately, terminologies here can be confusing. I would call $n$ the "exponent". But one might speak of $b$ as "$a$ to the power $n$" (which makes it sound like $n$ is the "power"), or "the $n$'th power of $a$" (which makes it sound like $b$ is the "power").

MAIN MEANINGThe difference between exponent and power can be drawn clearly on the following grounds: Quantity, describing the power to which base number is raised, is known as the exponent. The product of continuous multiplication of the same base number is called power.BASIS FOR COMPARISON EXPONENT POWERMeaningExponent means a quantity, describing the power to which base number is raised.Power implies the product of continuous multiplication of the same base number.RepresentsExponents : The number of times, the base number are to be multiplied together.Power : It represents two things, i.e. base number and exponent.In context|mathematics|language terms the difference between exponent and power is that exponent is (mathematics) the power to which a number, symbol or expression is to be raised for example, the 3 in x 3 while power is (mathematics) a product of equal factors notation and usage: x''''n'', read as "''x'' to the power of ''n''" or "''x'' to the ''n''th power", denotes ''x'' × ''x'' × ''x'', in which ''x'' appears ''n'' times, where ''n is called the exponent; the definition is extended to non-integer and complex exponents.hope it helps,follow methanks