Series of accumulated products.
Definition of powers
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Equivalence and commutive property of division. ||| Concepts and Numbers. ||| Bend coefficient of curves ||| Mathematical dimensions
Transposition property ||| Accumulated product: Powers ||| Dimensional Geometry: Reversibility
Priority Rule in powers and roots ||| The decimal counter ||| The floating point index ||| Paradoxes in mathematics
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Chaos Fecundity. Symbiosis: from the Chaos to the Evolution.||| Speed-Chords in galaxies.
The man and the testosterone.||| Toros say
Who is God
Series of accumulated products
Definition of Power.
Product between two numbers
The product between two numbers (a x b) will be the result of adding the value of (a) so much times as units (b) has.
Although to multiply, we use tables (memorized) that allow us the direct multiplication without necessity of making the sequential addition, which on the other hand, allows us the multiplication of decimals, fractions, etc.
Accumulated Product will be the product applied to the result of anterior multiplication.
For example, given 3x5x6, where firstly we make the multiplication 3x5 = 15 and later on we make the multiplication 15x6 = 90
The accumulate product have the particularity of been made step by step when not having tables that would allows us to make in a unique operation.
For example, the accumulated product 3x5x6, as we have seen, is not possible to make in alone one operation, but in two accumulative operations.
Series of accumulated products:
Series of Accumulated Products will be when we multiply all the terms (a,b,c, ....) of a series (set or succession) in the order of representation.
For example, the series 3x5x4x2 will be resolved: 3x5 = 15; 15x4 = 60; 60x2 = 120
In the series the total quantity of terms to multiply is defined as degree (n) of the series.
In the anterior example, n=4 mean to be a series of fourth (4) degree.
Powering series will be when all the terms of the series of accumulated products are equal.
For example, given de series 3x3x3x3x3 when having equal all the terms, it will be a powering series.
In the powering series the first and second terms will also be considered accumulated products by convention.
The powering series can be simplified by mean of a base (a) that represent the repeated term, and by mean of a exponent that represent the degree (n) of the powering series.
To this simplification of the powering series we can name as Power.
For example, given a powering series 3x3x3x3x3, where its simplification o power is 3^5.
Definition of Power :
As for the anterior considerations, the definition of power could be:
"Power is the accumulated product of a series of equal terms."
Very often we find in power form many operation that are not powers really, as in the case of roots.
In the case of a square root, we also can express it in power form with a base (a) and an exponent (1/n), as for example 9^(1/2)
But in this case it is not a power exactly, but its inverse operation, the root; and for this reason we also have to change the sense of seeking, say, here we have the accumulated product (9) of a series of degree (2) and what we have to find is the base (a) that is the root result.
In the case of 9^(1/2) = 3; the resulting 3 will be the base (a) of the accumulated product of the series (9) that was known.
Say, in the roots we seek the base of the accumulated product of the series.
Definition of root
As for the anterior considerations, the definition of roots could be:
"Root of a given accumulated product is and consists in obtaining the base (a) of the series that gives us that accumulated product."
It could occur also that we could have an exponent in form of fraction, as for example 4^(3/2).
In this case the operation would be double because we have a accumulated product and two degrees (n,m) to resolve.
Firstly we adjust the accumulated product of (n) degree, and later on with this partial result we proceed to solve the base of the (m) degree.
For example, given 4^(3/2), where firstly we adjust the (n) degree 4^3 = 64 as accumulated product, and later on we adjust the (m) degree as root 64^(1/2) = 8
Methods of powers and roots.
Direct or priority power -3^4, and series power (-3)^4
Direct or priority power will be when the power is applied uniquely on the numeric module, and this way, the module is revolve firstly and later on we apply the sign that the base got.
For example, given -3^4 = -(3x3x3x3) = -81
It will be complete power when we apply the power jointly to the numeric module and sign.
For example, given (-3)^4 = (-3x-3x-3x-3) = +81
The direct or modular root will when the root is applied alone on the module of the base, and later on, we apply the sign that this base got.
For example, given -81^1/4 = -(81)^1/4 = -3
Complete root will be when the root is applied jointly to module and sign, given in this case imaginary results with even root of negative number.
For example, given (-81)^1/4 = 3i