Modular or Priority Rule in Powers and Roots
(sg.X)Q = sg.|X|Q Where sg is the sign +/-
Natural, physical and real rule
In the physical and real world it is not found any negative value or potential that is converted into positive when being elevated to the square; neither even natural roots of negative numbers could change to imaginary numbers.
This way, it is necessary a rule to give solution to this real problem.
Solutions to powers and roots: Simple form -9 1/2 = -3 and composed or complex form (-9)1/2 = 3i

Of ferman: Fernando Mancebo Rodriguez--- Personal page.

"" The simple or real solutions to powers and roots of negative numbers are also, by natural logic, negative numbers.""
Rule that allows us operations of powers and roots without abandoning the negative field.  You can see many of my works, in the following pages:

Email: ferman25@hotmail.com
Email: ferman30@yahoo.es

Modular or Priority Rule in powers and roots.
"" The signs' rule is not suitable in simple solutions to powers and roots "".
"Because a single sign cannot operate on itself, but creating compositions with other signs."   "For any power or root a modular resolution or Priority Rule exists, for which, we solve firstly the power or root of the numeric modulus |absolute value|=(base or radicand without sign) and later on we apply the sign that the numeric modulus took"
Because to a single sign, e.g. ( - ), we cannot elevate it to powers neither to find it roots, but composing it with other signs only.

The priority rule says us that any pure power and root has to be resolved by mean of the following equality:
Given an equality (sg.x)n = sg.|x|n where sg. is the sign. +/- Conceptual difference among simple or natural powers and roots, and composed ones.

Conceptual difference between powers and roots of ONE negative number,
and composed products and roots for SEVERAL negative numbers.

-- In ONE negative number alone a sign exists, and therefore the powers and roots must to maintain in the same negative field.
*Any power and roots should be resolved by this simple procedure.

-- In the product and roots for SEVERAL negative number, also several negative signs exist and then is necessary the application of the method of composition of signs.
* In this case we must to take conscience that we are treating with not simple or natural powers and roots, but with multiple product (multiplication) and roots (decomposition), which can produce change of sign and field. (negative to positive, or vice versa.)

This way in this theory, the current form (-6)2 = 36 is not a direct powers, but a product composed by two negative terms (-6).(-6) = 36, where de operative system (signs rule) go changing the nature and field of numbers (negative/positive alternatively).
Contrary, the direct form of the Priority Rule is solved as: (-6)2 = -|6|2 = -36, where the priority is on respecting the nature and field (negative) of the numbers.
For example: (-5)2 degrees = -25 degrees Postulate:

The alleged mathematical differentiation between "a negative number" and "minus a positive number" has no consistency as is easily shown that both means the same thing:
For example:
[-5] and [- (+5)] has the same value and mathematical sense because both have the same value - 5, or what is said, a negative number.
This case, the expressions 6-2 = 6 - (+2) = 6 + ( -2) = 4 are the same thing, say, that we subtract 2 units from the number 6, in such a way that the expression -2 or (-2) is always a negative number because of always represents a subtraction or negative value for any count. Therefore what varies on solutions to power and roots are the mathematical methods applied to the same values.
In the method of signs you should use parenthesis to define that we want to compose and change signs.
(-3)2 = + 9 ----- (-9)(1/2) = 3i
In the method of priority or modulus rule we don't use brackets because we want to respect the sign field (+/-) where we are, and operate only on the module value.
-32 = -9 ------ -91/2 = -3 Change of sign procedure

I understand that with the change process of signs, it is demonstrated that is not equal the power of a negative number than the multiplication of several equal negative numbers.
For example:
Given (x)2 = -16, we can interchange of side the members, or change their signs without brake the equality.

(x)2 = -16 -------------> -(x)2 = 16
-x = 16 1/2 ----> -x = 4 ---> x = -4

Contrarily, in the multiplication of several equal negative numbers the equality doesn't have real solution. (x) * (x) = -16

Philosophy and nature of the Priority Rule:

Any number or quantity that is subjected to powers or roots should be considered as an internal operation on itself, and therefore, it must conserve its same nature and sign. ( Conservation of field or state.)
Otherwise it would not be operating on itself, but rather with other abstract numeric value that changes its peculiarities or operative field.
Therefore, when by mean of the sign's rule we force to a sign to operate on itself, the result can move away from the reality and mathematical logic, as in even powers and roots of negative numbers (- . - = + ) So, the Priority Rule gives us simple and real solutions to even powers and roots of negative number, maintaining the same sign and same operative sense.
And this way, this rule opens us the field of the real numbers to operations of powers and roots with negative numbers to which we didn't have access with the sign's rule. * To deny the existence and possibility of internal operations of even powers and roots inside the field of the negative numbers (and without being left this field) it is unreal and unacceptable mathematically.
So, to say that -2 is the fourth root of -16 inside the field of the negative numbers and without leaving this field, it is certain and inappealable, and for that reason we must to be able a mathematical norm to solve these circumstances: The Priority Rule.
-(e. g. square root of -25 euros = -5 euros / Square of -8 euros = -64 euros.) Then we can say:
The direct or priority square root of -16 degrees is -4 degrees, but never +4i degrees.
The direct or priority square root of -64 euros is -8 euros, but never +8i euros.
The direct or priority square root of -25 apples is -5 apples, but never...
The direct or priority square root of a acceleration of -49 km/s. is -7 km/.s, but never...
The direct or priority square root of -81 is -9. but never...
So, roots of negative numbers give us also negative numbers. ** To understand a little better the foundation of the Priority Rule, I put a synoptic scheme of the fields and groups in that we can frame the numbers. In it, you can see as any number is defined by two parameters: Its characteristic group and its field that can positive or negative. Operative ways: scheme

Firstly, we can see the two forms of expressing the powers and roots: in its simple or priority form, where we apply the powers and roots alone on the numeric modulus; and in its composed form, where we apply the powers and roots to the modulus and sign together. In the following drawing we see as starting from any composed power or root, we can substitute the incognita (x) and to be transformed into simple real or priority solution xp. Real and abstract meaning in powers and roots:

In powers with real meaning (priority rule) we multiply the absolute value of a number (or quantity, applicable to objects) by itself N times, but conserving its sign or field, either positive or negative.
-32 euros = -|3|2 euros = - 9 euros.

On the other hand the abstract powers (N) (sign rule) what is made is to multiply the base (that can be applicable to a quantity of objects) by (N-1) abstract bases, those which can change the field or sign.
(-3)2 euros = |- 3 euros| x (- 3) = 9 euros. Logical, primitive and natural thought:
"The square of something negative has to be something more negative yet"

Definition ambit of the negative powers and roots:
During the time that a real number is defined or it acts as negative quantity, their powers and roots are also real and negative

This way in powers and roots:
1.- If we want to maintain us in a real ambit, we should apply the priority rule.
---- This case, we subject the operative method to the number qualities
2.- If we want to enter in an imaginary, complex or abstract ambit, we can use the sign's rule.
---- This case, we subject the number qualities to the operative method. Signs Rule: Simple analysis

On the other hand, we can prove that the signs rule is wrong from its principles and postulates.
The signs rule says us the even powers of a negative number is equal to the same power of that positive number, say (-x)2 = x2
But this is false because if we make the square root to both terms, the result is -x = x, being false.
If we resolve -x, then -x = (x2)1/2 being this way -x = x, also false.
If we resolve x, then x = [(-x) 2]1/2 being this way x = -x, also false.
So, the basic and fundamental equality (-x)2 = x2 of even powers and roots of negative numbers is false from its beginning.
This way, if we would that this equality has to be possible we must express it in form of product of numbers. (-x)*(-x) = x2.

In the same sense, in the current theory is not completed the Inversion proof for powers and roots, which is correct for the Priority Rule. Preamble

A first reminder drawing For the resolution of operations in such series as polynomials, we have chosen a specific priority system (or direct method) for each class of operations, in such a way that we begin solving firstly the terms with powers and roots; later the products and divisions, and finally, we give solutions to the operations of addition and subtraction.

1.- Powers and roots.
2.- Products and divisions.

Besides, we have settled down an operability norm with the signs + and - ( Signs' rule - abstract or complex method ) such that:

+ by + = +
- by - = +
+ by - = -
- by + = -

* by = Composed with, combined with.

Nevertheless I think that in some operations as powers and roots the solutions are not completely gotten when we use the complex or abstract signs' rule, and in some cases they are quite illogical, as for example when we say with negative numbers the even powers are positive and the odd ones are negative.
Logically the powers should have the same sign, since for example of the step from power 2 to power 3 alone the exponent increase, but any other sign change doesn't exist.

This way in the powers, we use an indirect resolution method (Subtraction from the subtrahend) based on successive multiplications:
-34 = (-3) x (-3) x (-3) x (-3) = 81
by means of the signs' rule, which doesn't seem to be correct for the mathematical reality that doesn't have to complete this sequence of products with alternation of signs ( flaw of method? ), but rather it should be a direct resolution and relation between power and result:
|-34| = |-81|
That is to say, to proceed to the numeric power and conservation of the same quality or sign.

The sign alternation in the powers of negative numbers doesn't happen in the applied reality, it is simply an abstract state.
(e.g.) The square of -7 degrees in a thermometer will be -49 degrees, but never + 49 degrees.

Equally it happens with roots, since we say the even root of negative number doesn't have real solution. And it should have it.

We can put as example to a thermometer that marks us -81 degrees.
Those -81 degrees yes, they HAVE its corresponding real fourth root that is -3 degrees. ( -811/4 = -3, since -34 = -81 ).
But we cannot obtain it by means of the sign rule that gives us a complex solution, but by means of the priority rule that gives us a practical and real solution. Therefore, and as preamble, we can say:
"The sign's rule is an abstract method, sometimes contradictory and not adequate in some operations as powers and roots for the resolution of real situations, in which, we should apply the priority rule."

* And let us remember:
When an operative method doesn't coincide with the reality, it is the method that is incorrect, but not the reality. And it is with the intention of to solve or improve this problem, for what I propose the priority solution that is not an arbitrary method, but simply to follow the operative norms already accepted and before mentioned.

Development Powers: "The power of a negative number is always a negative number" Conservation of field or state

As we have said, the resolution system in polynomials it is the one of establishing an operative priority with the following sequence:

1.- Powers and roots.
2.- Products and divisions.

Nevertheless, many time will have some circumstances and solutions such as the following one: (1)

36 - 32 = 27
- 32 = 27 - 36
- 32 = - 9

Although, this equality is not accepted at the moment.
And the question arises: is it correct or not this solution?
Naturally I understand that yes, this solution is correct following the priority norm in the resolution of these operations.
Let us see:
The equality 36 - 32 = 27 is true whenever in the second term o monomial ( - 32 ) we proceed to operate following the priority procedure, that is to say, to solve the power firstly and later to apply the subtraction sign.

- 32 = - ( 3 x 3 ) = - 9 Priority solution. But to proceed in contrary sense (as we make at the moment) we should expose it in the following form:

(- 3) 2 = ( -3 ) x ( -3 ) = 9 Therefore the direct, correct and high-priority solution will be the first one: - 32 = - ( 3 x 3 ) = - 9 , the one that is deduced in the resolution ( 1 ).

It is also the more mathematical logical solution, because it seems unacceptable, as we said at the beginning, that in a series of solutions of powers of negative numbers, the even powers are positive and negative the odd powers.

This way given an real number -2 that has powers -21 = -2 ; -23 = -8 ; -25 = - 32 ; etc., Also it has to have intermediate powers that give us -22 = -4 ; -24 = -16 etc.

And the procedure to get this is alone to follow the priority rule in the polynomials resolution. Then, the way to express and resolve powers by means of the priority rule or by means of the sign's rule would be: Roots: "The root of a negative number is always a negative number" Conservation of field or state

For the same principle of priority, to the roots we can give them the same solution that to the powers:
To make the root operation firstly on the radicand module (without sign) and later to apply it the notation sign + or - that takes the radicand.

Given a square root (R2/) of negative radicand -16, R2/-16 in which we solve firstly the root of the radicand (without sign) 16, and later on, we apply it the sign that possesses this radicand.

R2/ -16 = - (R2/16) = - 4

All that due to the reciprocity between roots and powers:
"The result of any N-root of radicand ( a ) will be the number that elevated to N give us the radicand ( a )."

This way, if we have a root R2/-16, its result will be the number that elevated to the square give us -16. To know: - 4

- 42 = - 16

To see it a little easier, we will use the exponential form of representing of roots: b1/n
Let us give an example.
Given a square root: -161/2

-161/2 = -( 161/2 ) = -4 In this case, we see more clearly that the numeric module (without sign) can, and it has priority, to be solved firstly and later on to be applied the sign ( - ) that took this module.

Otherwise we would have to express it this way: ( -16 )1/2 = -4i

Historical confusion:
The historical confusion that has impeded the development, concretion and correct application of the priority rule has been consequence (I think) of the use for roots of its specific symbol or radical sign |/, that contains inside it the sign and the radicand, which seems to force to make the operation by mean of the sign rule.
But it is alone a subjective appreciation taken place by its graphic form, but not real.
However when we expose a root in exponential form, then we see clearer the application of the priority rule, as in any other exponent.
Therefore, and as we have seen, the priority rule is acceptable and applicable en any case. Example of solution of roots by mean of the priority rule: How to write the wanted roots? Therefore, with the rule of priority we solve, or we give an acceptable solution, to the negative powers and roots.
And this solution seems to be the most logic in practice since continually we have a lot of problems that need this type of solution "of priority".
For example,

1. - If a thermometer marks -3 degrees, and you want to elevate this value to the square, the result will be of -9 degrees, but not of +9 degrees.
If now with -9 degrees you want to take it until its square root, then the result is of -3 degrees again, but not of an imaginary number 3i.

2.- If on some Cartesian coordinates we have a value of -4x, when elevating it to the square we will also have a negative value of -16x, but not of +16x.
If now to this value of -16x we want to calculate its square root, the result is of -4x again, but not an imaginary value of 4ix.

3.- If on the level of the sea (0) we measure 5 meters deep, (-5 meters on the level) and we want to elevate it to the square, the result will be of -25 meters on the level of the sea, but not +25 meters.
If now we want to find the square root of these -25 meters, the result will be of -5 meters, but not 5i meters.

4.- If a bank gave me a credit of 160 euros, my personal account in this bank will be of -160 euros that I must to pay next.
Although, if en any moment I desire elevate to the square this credit and the bank accepts, then I have an account of -25600 euros that I must to go paying en the future. But, I have not an account with +25600 that the bank has to give.
So in this case, the square of these -160 euros of credit (-1602) will be -25600 euros of credit.

But if I desire to know what will be the square root of my credit (-2500 euros) then I can adjust and see that this squared root is of -50 euros, but not an imaginary number 50i.

In the same sense, we should keep in mind that the sign minus ( - ) it is the sign of the subtraction or subtrahend, and in any subtraction we can elevate to the square to the subtrahend without for it this sign changes and become + or addition. Example of problem with priority rule solution:

If we have 1 litre of water to -16 degrees and other one to 16 degrees, and we mix them, we obtain 2 litres of water to 0 degrees.
This way, the question is: what would the resulting temperature be?
- 16 x 1 + 16 x 1 = - 16 + 16 = 0
If now we want to make the same experiment with the fourth root of these temperatures, which would the adjustment be?.
-161/4 x 1 + 161/4 x 1 = - 2 + 2 = 0 Examples of operations with priority rule:  In general, the Priority Rule means that:

"On a value or quantity taken toward certain direction, we can apply any operation of powers and roots, without for it this value changes its direction or sense ( +, - )."

This way, the priority rule allows us to operate in a practical way on real values, avoiding the changes of sign and meaning, to which, the abstract application of the signs' rule could take us.

So:

The even powers or roots of negative numbers (by mean of the "abstract" signs' rule) can give us positive or imaginary results, but its "practice and substantial" powers and roots always give us a negative real number (priority rule).

* On the other hand, the priority rule is already applied in the simplest operations as a sum of negative numbers would be:
-3 -4 -6 -9 = -22
which is solved adding the numeric modules firtly 3 + 4 + 6 + 9 = 22, and later we put the minus sign (-) that the numbers took (- 22).
The same circumstance that the priority rule on powers and roots.

For it to these solutions for negative powers and roots will denominate them "solutions of priority"; and to this rule (that is used already commonly as resolution of polynomials), we will denominate "Priority Rule."

On the other hand, the Priority Rule solves us with real numbers many applications that needed imaginary numbers, as those that we have seen previously and in the following drawing.
And also solutions for operations that now they don't have: x4 + 16 = 0 Therefore and to finish, let us make a first definition intent and let us expose that:

"For any power or root a direct resolution or Priority Rule exists for which we solve firstly the numeric module (base or radicand without sign) and later on we apply the sign that the numeric module took"

This way, we could say that:

"In practices, the operative reality demands the existence and application of the Priority Rule." Practical and operative meaning

This way when we have to solve a power or root in a practical or real way, as for instance the square of -3 degrees, we can take two different positionings:
--- One, to use the abstract or indirect method following the signs rule, where we multiply the -3 degrees by the abstract number (-3), and obtaining a complex product with change of sign or numeric field. (-3g.) x (-3) = + 9g.

--- Two, using the priority rule y or direct power, in which we elevate to the square the -3 degrees, with sign conservation and field conservation: -3g2 = -|3|2g = -9g.
As we see the rule of the signs gives us a complex mathematical product and the rule of priority gives us a real square (physical square). Abstract and real: Two operative methods.  Thank you. Ferman: Fernando Mancebo Rodriguez 2010-12-12