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- Exponentiation

**Exponentiation** is a mathematical operation, written ** a^{n}**, involving two numbers, the

*a*^{n}=*\underbrace{a* x … x *a}*_{n,}

just as multiplication by a positive integer corresponds to repeated addition:

*a* x *n*=*\underbrace{a*+ … +*a}*_{n.}

The power *a*^{n} can be defined also when *n* is a negative integer, at least for nonzero *a*. No natural extension to all real *a* and *n* exists,but when the base *a* is a positive real number, *a*^{n} can be defined for all real and even complex exponents *n* via the exponential function *e*^{z}. 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 as well, 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 exponentiation operation with integer exponents requires only elementary algebra.

The expression *a*^{2} = *a*·*a* is called the square of *a* because the area of a square with side-length *a* is *a*^{2}.

The expression*a*^{3} = *a*·*a*·*a* is called the cube, because the volume of a cube with side-length *a* is *a*^{3}.

So 3^{2} is pronounced "three squared", and 2^{3} is "two cubed".

The exponent says how many copies of the base are multiplied together. For example, 3^{5} = 3·3·3·3·3 = 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*, and 243 is the *power* or, more specifically, *the fifth power of 3*, *3 raised to the fifth power*, or *3 to the power of 5*.

The word "raised" is usually omitted, and very often "power" as well, so 3^{5} is typically pronounced "three to the fifth" or "three to the five".

Formally, powers with positive integer exponents may be defined by the initial condition *a*^{1} = *a* and the recurrence relation *a*^{n+1} = *a*·*a*^{n}.

Notice that 3^{1} is the product of only one 3, which is evidently 3.

Also note that 3^{5} = 3·3^{4}. Also 3^{4} = 3·3^{3}. Continuing this trend, we should have

3^{1} = 3·3^{0}.Another way of saying this is that when *n*, *m*, and *n* − *m* are positive (and if *x* is not equal to zero), one can see by counting the number of occurrences of *x* that

x^{n} | |

x^{m} |

=*x*^{n}*.*

1=

x^{n} | |

x^{n} |

=*x*^{n}=*x*^{0}

Therefore we **define** 3^{0} = 1 so that the above equality holds. This leads to the following rule:

- Any number to the power 1 is itself.
- Any nonzero number to the power 0 is 1; one interpretation of these powers is as empty products. The case of 0
^{0}is discussed below.

For nonnegative integers *n* and *m*, the power *n*^{m} equals the cardinality of the set of *m*-tuples from an *n*-element set, or the number of *m*-letter words from an *n*-letter alphabet.

0^{5} = |