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Fourth Sursolids, or 13th Pow'r 1|| 8192 1594323 67108864 1220703125 13060694016 96889010407 549755813888
2d Sursolids Sqd. or 14th Pow'r 1 16381 4782969 268435456 6103515625 78364164096 678223072849 4898046511104 22876792454961 Sursolids Cubed, or 15th Pow'r 1 32768 14348907 1073741824 30517578125 470184984576 474756150994335184372088832 205891132094649
There is no number of which we cannot find any power exctly; but there are many numbers, of which the exact roots can never be obtained. Yet, by the help of decimals, we can obtain these roots to any necessary degree of exactness.
Those roots which cannot be exactly obtained, are called surd roots; and those which can be found exactly, are called rational roots.
Roots are sometimes denoted by writing this character before the power, with the index of the power over it; thus the cube root of 64 is expressed 64, and the square root of 64 is expressed 64, the index 2 being omitted when the square root is required.
EXTRACTION OF THE SQUARE ROOT.
THE EXTRACTION OF THE SQUARE ROOT is the method of finding a number, which, being multiplied by itself, shall produce the given number.
1. Distinguish the given number into periods of two figures each, by putting a point over the place of units, another over the place of hundreds, and so on, which points show the number of figures the root will consist of.
2. Find the greatest square number in the first, or left hand period; place the root of it at the right hand of the given number, (after the manner of a quotient in division,) for the first figure of the root, and the square number under the period, then subtract it therefrom, and to the remainder bring down the next period for a dividend.
3. Place double of the root, already found, on the left hand of the dividend for a divisor.
4. Seek how often the divisor is contained in the dividend, (except the right hand figure,) and place the answer in the root for the second figure of it, and likewise on the right hand of the divisor; multiply the divisor with the figure last annexed by the figure last placed in the root, and subtract the product from the dividend to the remainder join the next period for a new dividend.
286. Can you find the power and root of any number? 287. What is the distinction in the roots?— -288. What is the extraction of the Square Root?—289. What is the rule?
5. Double the figures already found in the root, for a new divisor, (or bring down your last divisor for a new one, doubling the right hand figure of it,) and from these, find the next figure of the root as last directed, and continue the operation in the same manner, till you have brought down all the periods.*
The rule for the extraction of the square root may be illustrated by attending to the process by which any number is raised to the square. The several products of the multiplication are to be kept separate, as in the proof of the rule for multiplication of simple numbers. Let 37 be the number to be raised to the square.
Now it is evident that 9, in the place of hundredths, is the greatest square in this product; put its root, 3, in the quotient, and 900 is taken from the product. The next products are 21421=2×3×7, for a dividend. Double the root already found, and it is 2X3, for a divisor, which gives 7 for the quotient, which annexed to the divisor, and the whole then multiplied by it, gives 2×3×7(42)+7×7(=49) which, placed in their proper places, completely exhausts the remainder of the square. The same may be shown in any other case, and the rule becomes obvious.
Perhaps the following method may be considered more simple and plain. Let 37=30+7, be multiplied as in the demonstration of multiplication of simple numbers, and the products kept separate.
The root of 900 is 30, and leaves the two other terms, which are exhausted by a divisor formed and multiplied as directed in the rule.
290. Explain to me the nature of this rule.
NOTE 1. If when the given power is pointed off as the power requires, the left hand period should be deficient, it must nevertheless stand as the first period.
2. If there be decimals in the given number, it must be pointed both ways from the place of units: If when there are integers, the first period in the decimals be deficient, it may be completed by annexing so many ciphers as the power requires: And the root must be made to consist of so many whole numbers and decimals as there are periods belonging to each; and when the periods belonging to the given number are exhausted, the operation may be continued at pleasure by annexing ciphers.
1. Required the square root of 729 ?
729(27 the root.
The given number being distinguished into periods, we seek the greatest square number in the left hand period (7) which is 4, of which the root (2) being placed to the right hand of the given number, after the manner of a quotient, and the square number (4) subtracted from the period (7) to the remainder (3) we bring down the next period (29) making for a dividend, 329. Then the double of the root (4) being placed to the left hand for a divisor, we say how often 4 in 32? (excepting 9 the right hand figure) the answer is 7, which we place in the root for the second figure of it, and also to the right hand of the divisor; then multiplying the divisor thus increased by the figure (7) last obtained in the root, we place the product underneath the dividend, and subtract it therefrom, and the work is done.
DEMONSTRATION OF THE REASON AND NATURE OF THE RULE.
The superficial content of any thing, that is to say, the number of square feet, inches, &c. in the surface of a field, a floor, &c. is found by multiplying the length into the breadth. Thus, if a piece of land be 10 rods in length, and 10 in width, it is a square, and the measure of one of its sides is the root, of which the superficial content of the piece of land is the 2d power. Or, supposing you have a piece of cloth 1 yard wide, and 225 yards in length, and you wish to know how many square yards it will cover, you must so arrange the parts of the whole that they may be in a square form.
Now, suppose you have 144 square pieces of wood, and wish to know how many pieces would be on a side, were the whole arranged into a square form. To determine this, you must extract the square root 144; the first step of which is to point off the number into periods of two figures each. This shows how many figures the root will consist of, and is done on this principle, that the product of any two numbers can have, at most, but as many places of figures, as are in both the factors, and at least but one less.
The left hand period being 1, the square of it will be 1, and likewise the root will be 1. But as we have nothing to do at present with the right hand period, we will omit it, and consider only the left hand period, which being in the place of hundreds, must be called 100; hence the operation, at present, will be to find the square root of 100. The root of 1, is 1, but as there are two periods in 100, there will be two figures in its root, and as the figure already obtained in the root is equal to its period, there is nothing remaining for the next period; and as the next period consists wholly of ciphers, the next figure of the root will be a cipher, so that the root of 100 is 10. By this process we have disposed of 100 of the pieces into
the form represented by Fig. 1, viz. 10 pieces on a side.
The reason for placing the square number underneath the period, and subtracting it from the period, as directed in the rule, is as follows. When we have obtained the root of the left hand period, we have disposed of as many pieces as the greatest square of the left hand period represents, and by subtracting the square of the root from its period, we make it smaller by as many as the square of the root represents; thus in the example given, 1 in the quotient represents 10, the square of which is 100, which 1, under the left hand period, represents. This, subtracted from the left hand period, leaves 44; so that 100 pieces have been disposed of as represented by Fig. 1, and 44 pieces are now to be added to it, in such manner that the square form will be preserved. To do this, the rule directs to place the double of the root already found on the left hand of the dividend for a divisor."