What is the difference between exponent, power and index?

Question

I have read that, if $$b=a\times a\times a\times \cdots \times a \text{ ($n$ times})$$ then $$b=a^n$$ where $b$ is known as the base, $n$ as the index or exponent and $b$ is the power. The author in many places refers to both $b$ and $n$ as powers. Is this correct? Is there any difference between exponent, index and power?

Answer 1

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").

Answer 2

Most sources would go with the model in the picture below, however many of those sources lack an unambiguous explanation. The "power" is not a "visible thing" more than it is a way of saying what the exponent is doing with the base. So in the image there are only two things, the exponent and the base. Together they form a power.

Image