Obtaining powers and roots in mathematics today are easily executed by pressing the correct keys on an electronic calculator. But, one can only wonder hoe the ancient peoples obtained such omputations. The ancient Babylonians and Egyptians seem to have had computational methods based on simple addition and division. One of the main methods employed was that of mediatio/duplatio, halving and doubling the numbers. We have used this same method throughout our research in an attempt to reconstruct some of the possible computations. In this Extract (N°. 4), we shall illustrate some of the possible methods that one could use to obtain certain products of powers and roots through simply adding and dividing numbers based on the concepts of doubling/halving numbers.

In earlier essays, we have seen how the maya long count numbers and the numbers of the ancient kemi system are apparently related. We shall use numbers that are readily accepted as being significant for those two ancient cultures. In reviewing the distinct examples of the Extract, one should keep in mind the concept of fractal numbers, whereby it is significant to visualize 4 x 4 = 16, as also being 4 x 40; 40 x 40; 40 x 400; etc.

POWERS

The square of a particular number may be easily obtained throughout contemporary methods of multiplication. In fact, there are alternative ways of achieving this; consider the 52c numbers of the ancient Mesoamerican system:

208² = 43264

200 x 200 =	40000
400 x 8 =	3200
8 x 8 =	64
43264

In these cases, one simply takes the initial hundreds of the number under study (200, or 400 here) and multiply them as fractals and adding zeros. Then, the second step of computation doubles that particular number times the last two digits of the number studied; and, finally, the third step consists of multiplying the last two digits times themselves. Then, those products are added together for the final result. Consider another example:

These steps in multiplication may be achieved through addition as well; we have made the presentation more brief.

Cube roots present another procedure which involves the maya long count numbers directly. Let us look at some examples of the same 52c series and the corresponding power of three.

In both of the cases above, the same integers of the numbers themselves (208, 416) are used for the computations, except for the fact, now, in powers of three for the 52c numbers, the maya long count numbers are used for obtaining the final result. Both 576 and 4608 are numbers within the maya long count and can be employed for obtaining the final results as shown above through subtraction. The rule appears to be that if the number subtracted initially (512) is used, then one would employ the maya long count number nearest it: i.e., 576 in this case; whereby 4608 is the nearest number for the 4096 constant number. One must also understand that the numbers 512 and 4096 lie on the series of constant numbers: 2,4,8,16,32,64,128,512,1024,2048,4096

In powers of three, then, significantly appear the three distinct series of numbers in relation to one another in simple computations of addition and subtraction:

13,26,52,104,208,416,832,1664...
18,36,72,144,288,576,1152,2304,4608,9216...
2,4,8,16,32,64,128,256,512,1024,2048,4096...

Once again, it becomes obvious that the selection of the 52c, the 360c, in relation to the constant numbers in relation to the power of three surely could not have been made through accident in ancient times. The ancient peoples just happenend to employ three series of numbers, which just happen to produce computations for the power of three. Obviously this was no accident. Let us observe the procedure for the 54c of the ancient kemi in Egypt:

In this manner, we also see how the 108c (54c) of the ancient kemi connects easily to the maya long count numbers for obtaining products to the power of three. Even complex looking numbers like
1664³ ( = 4607442944) are easily computed: 1600³ = 4096000000, plus 800³ = 512,000,000 yielding a maya number 4608000000, then has the 64³ = 262144 subtracted from it:
4608000000 - 262144 = 4607737856. Then, the maya number immediately greater than that 294912 is thereby subtracted from that: 4607737856 - 294912 = 4607442944. In other words: 1664³. This seven-step procedure makes computations for the power of three extremely easy for the 52c and the 54c of the ancient kemi-maya system.

ROOTS

A discussion of powers cannot be complete without examples for obtaining the roots of equally complex numbers. For our example, we shall use a number found in the ancient kemi system: 756, which is often cited as the base measurement of a side of the Great Pyramid. Let us say, we know the square of 756 to be 571536. Now, let us wish to computate the square root of a related number, for example, 571541. Without the assistance of the electronic calculator these computations become almost unmanageable. However, let us explore the idea of a constant number as of the 571536 value. Consider the following method.

we shall now take this constant value (at .000661375) and add it to 756 in an incremental manner in order to obtain the following roots and powers after 756²:

Other results may be easily obtained by use of the constant value:

In order to make adjustments and eliminate the decimal fractions in these results, one would simply reduce the decimal places used in the computations; consider,

The electronic calculator offers the number 756.999339497 for the rot of the number 573048; a difference of about .000093; something next to an incommensurable distinction.

Now, supposing one might wish to know the square root of 571541.5 of our example. Then, simply use 189 / 571536 = .0003306878307 as the constant value (189 being half of 378 which was employed above initially):

which is an extremely close approximation. The electronic calculator offers the value 756.003637557; a difference of .000000005; almost nothing to speak about.

Understandably, the electronic calculator must be computing numbers on a similar basis of constant values. However, what we find to be of interest is the manner in which the series of historically significant numbers relate to one another in terms of powers and roots. For example, consider some historically significant numbers of the maya long count: 1872000 and 151840. Other numbers may also be considered, but for now we shall exemplify these:

The number 1872000 (days) constitutes the cut-off period for the maya long count and reflects a significance for powers and roots:

If we take .2105 to be the constant value for the series as it is, then:

The terminations of the numbers in the series reflect the maya long count fractals: 576, 1152, 2304..., whereby the termination 728 (in 1873728) reflects the 1728 integers (that is 1152 + 576 = 1728).

The number 151840, another historically significant maya number often cited in the literature, reflects another series of powers/roots as of the number series 390c.

The constant value in this case would be .666524094; producing a constant series of numbers differentiated by the value of the 260c.

Other number series exist: for example, on the same maya series of 1872000, if one employs the kemi number 1296 in subtracting it from that maya period, the companion number 1366560 obtains:

There would be a corresponding constant value for each computational procedure for the powers of these numbers.