exponent
An exponent is a kind of shorthand for repeated multiplication. Instead of writing 2 × 2 × 2 × 2 × 2, you write 2^5 and read it as “two to the fifth power.” The small raised number, the 5, is the exponent; it counts how many copies of the base get multiplied together.
More precisely, in the expression b^n the base b is multiplied by itself n times when n is a positive whole number: b^n = b × b × ... × b with n factors. The exponent sits up and to the right of the base, and the whole thing is called a power. Exponents obey neat rules — adding when you multiply like bases, subtracting when you divide — that make working with very large or repeated products manageable.
Once n is allowed to be zero, negative, or fractional, the “repeated multiplication” picture stops working literally and the meaning is extended so the rules stay consistent: b^0 = 1, b^(-n) is a reciprocal, and b^(1/n) is a root. A common slip is to treat the exponent as a multiplier — 2^3 is 8, not 6, because it means 2 × 2 × 2, not 2 × 3.
3^4 = 3 × 3 × 3 × 3 = 81. Here the base is 3 and the exponent is 4.
The exponent 4 counts four factors of the base 3.
“Exponent,” “power,” and “index” are often used loosely for the same small raised number. In careful usage the exponent is the number, while the whole expression b^n is the power.