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Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent (or index or power) n. When n is a positive integer, exponentiation corresponds to repeated multiplication; in other words, a product of n factors, each of which is equal to b (the product itself can also be called power):
just as multiplication by a positive integer corresponds to repeated addition:
The exponent is usually shown as a superscript to the right of the base. The exponentiation bn can be read as: b raised to the n-th power, b raised to the power of n, or b raised by the exponent of n, most briefly as b to the n. Some exponents have their own pronunciation: for example, b2 is usually read as b squared and b3 as b cubed.
The power bn can be defined also when n is a negative integer, for nonzero b. No natural extension to all real b and n exists, but when the base b is a positive real number, bn can be defined for all real and even complex exponents n via the exponential function ez. Trigonometric functions can be expressed in terms of complex exponentiation.
Exponentiation where the exponent is a matrix is used for solving systems of linear differential equations.
Exponentiation is used pervasively in many other fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography.
The expression b2 = bA?b is called the square of b because the area of a square with side-length b is b2.
The expression b3 = bA?bA?b is called the cube, because the volume of a cube with side-length b is b3.
So 32 is pronounced "three squared", and 23 is "two cubed".
The exponent says how many copies of the base are multiplied together. For example, 35 = 3A?3A?3A?3A?3 = 243. The base 3 appears 5 times in the repeated multiplication, because the exponent is 5. Here, 3 is the base, 5 is the exponent, a

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