Empowerment

Empowerment . It is an operation that corresponds to the ordered pair (a, n) its power P. This being equal to the product of the factor a by itself n times. To abbreviate the writing, this factor is written and, in its upper right part, the number of times it is multiplied is placed. Since it is not something other than multiplication, this operation is always possible within the system of natural numbers .

Table of Contents

Summary

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1 Nomenclature and notation
- 1 Axiomatic definition
2 Reading
3 Power of exponent 0
4 Power of exponent 1
5 Negative exponent power
6 Product of powers of equal base
7 Ratio of powers of equal base
8 Power of a product
9 Power of a power
10 Base Powers 10
11 Rational Powers of Real Numbers
12 Real Powers of Real Numbers
13 Complex power of a complex number
14 See also
15 References and notes
16 Sources

Nomenclature and notation

The second power of a number is called a square, so 9 is the square of 3
The third power of a number is called a cube, so 8 is the cube of 2
From 4 onwards it is said fourth power, fifth power, etc.

Generalizing: a x a x a x a ……… .. n times is the nth power of a .

The number that is taken as a factor is called the base and the number that indicates the number of times to take it as a factor is called the exponent or degree . The operation is indicated like this:

2 ⁴, which means that 2 must be taken four times as a factor. In other words: 2 x 2 x 2 x 2 = 16, that is: 2 ⁴ = 16.

Generalizing: a ⁿ indicates that we must take the number a , n times as a factor. That is: a ⁿ = a x a x a x a ……… n times.

Axiomatic definition

a ⁰= 1
a ^{n + 1}= a ⁿ ^[one]

It has been defined by applying the principle of mathematical induction.

It is proved that a ¹ = a; indeed:

a ¹ = a ^{0 + 1} since 1 = 0 + 1

a ^{0 + 1} = a ⁰ a, by part (2) of the definition.

a ⁰ a = 1 (a) = a. Since a ⁰ = 1, part (1) of the definition.

Finally a ¹ = a

The properties are also fulfilled:

a ^{h + k}= a ^h a ^k
(a ^h) ^k = a ^hk . In all these cases a, n, h, k are natural numbers

Reading

In general, to read an elevation to power that is indicated the base is read, next it is raised to … and then the exponent is read.

Examples:

2 ⁵is read two raised to 5.
3 ⁷is read three raised to 7.
a ⁿis read a raised to n.

However, when the exponent is 2 or 3, the current is to read respectively squared or cubed. Or more briefly by deleting the word elevated, thus 5 ² ; 5 ³ reads five to the square , five to the cube .

In the case that the base is a letter, the custom is to simply read the letter and its exponent. Example: a ⁴ ; b ⁷ ; a ⁿ are read respectively: a through four; b to seven; a to jan.

Power of exponent 0

Every number (other than zero) raised to the zero power is equal to 1, that is, ⁰ = 1

However, in the set N = {0,1,2 …} of the natural numbers it is shown that 0 ⁰= 1 ^[2] [3]
As limit of a real function of real variable

If the function y = x ^{x is} defined on the interval (0, + ∞), the limit can be found by passing the exponential form y = e ^xlnx ; the limit of the exponent can be found by H’opital, for which we write xlnx = lnx / (1 / x), deriving in both members of the fraction, it results (1 / x) / (-1 / x ² ) = -x, whose limit when x tends to 0, is zero. Therefore the limit of y is reduced to the value e ⁰ = 1. Consequently, the limit of x ^x is 1, when x approaches zero from the right.

Power of exponent 1

Every power of exponent 1 is equal to the base, that is, a ¹ = a.

Example: 452 ¹= 452

Negative exponent power

A number raised to a negative exponent is equal to the inverse of the same expression but with a positive exponent:

Since there is no inverse of 0, there are no powers of 0 with a negative exponent.

Product of powers of the same base

The product of several powers of the same base is another power whose base is the common base and the exponent of the sum of the exponents of the factors.

to ^m . a ⁿ = a ^{m + n}

Example: 4 ³4 ⁵ = 4 ^{3 + 5} = 4 ⁸

Equal Powers Quotient

The quotient of dividing two powers of the same base is another power that has the common base and the exponent, the difference between the exponent of the dividend and the exponent of the divisor.

a ^m : a ⁿ = a ^{m – n}

Example: 2 ⁷: 2 ⁵ = 2 ^{7 – 5} = 2 ²

Power of a product

The power of a product is equal to the product of the factors raised each to the exponent of said power. That is, a power of base a; b of exponent n, is equal to the factor a raised an, multiplied by the factor b also raised an:

(a. b) ⁿ = a ⁿ . b ⁿ

Example: (5. 4) ³= 5 ³ . 4 ³

Power of a power

The power of a power of base a is equal to the power of base a and whose exponent is the product of both exponents (the same base and exponents are multiplied)

(a ^m ) ⁿ = a ^m. n

Example: (6 ⁴) ³ = 6 ³

Base Powers 10

In powers with base 10, the result will be the unit displaced as many positions as indicated by the absolute value of the exponent: to the left if the exponent is positive, or to the right if the exponent is negative.

Examples:

10 ^-4= 0.0001
10 ^-5= 0.00001
10 ⁴= 10,000
10 ⁵= 100,000

Rational powers of real numbers

If we assume that a is a positive real number and p / q is a positive rational number, with p and q being prime to each other, what does a ^{/ p} mean ^?

Real powers of real numbers

How to understand 2 ( ²⁵)? In other words, explain what a power whose base is 2 and its exponent the square root of 2 means. This number is the so-called Gelfand number.

Complex power of a complex number

By definition, whatever the complex numbers a other than 0 and b are, set to ^b = e ^{bLn a. Being e}^w = exp {wLne} = expx {w (1 + 2pi i k)}. But if the opposite is not stated, k = 0 will be taken, that is, e ^w = exp w.

1 ³^1/2= cos (2k 3 ^1/2 π) + isen (3 ^1/2 π)
5 ⁱ= e ^{2k π} (cos ln 5 + isen ln 5)
1 ^-i= e ^2kπ
i ⁱ= e ^{(2kπ -1/2)}
(3 -4i) ^{1 + i}= 5e ^{arctg4 / 3 + 2kπ} [cos (ln5-arctg4 / 3) + isen (ln5-arctg4 / 3)]