RULE Multiply the given number continually by itself, till the number of multiplications be one less than the index of the power to be found, and the last product will be the power required. OR THE EXTRACTION OF ROOTS. EVOLUTION, or the reverse of Involution, is the extraction or finding the roots of any given powers. The root is a number, whose continual multiplication into itself produces the power and is denominated the square, cube, biquadrate, or 2d, 3d, 4th, &c. power, equal to that power. Thus, 4 is the square root of 16, because 4X4=16; and 8 is the cube root of 27, because 3×3×3=27; and so on. Although there is no number of which we cannot find any power exactly, yet there are many numbers of which precise roots can never be determined; but by the help of decimals, we can approximate towards the root to any assigned degree of exactness. The roots, which approximate, are called surd roots; and those, which are perfectly accurate, are called rational roots. Roots are sometimes denoted by writing the character ✔ before the power, with the index of the root over it; thus, the 3d root of 36 is expressed, 36, and the second root of 36 is 36, the index 2 being omitted when the square root is designed. If the power be expressed by several numbers with the sign+ between them, a line is drawn from the top of the sign over all the parts of it; thus, the 3d root of 42+22 is 、 42+22, and the second root of 59-17 is /59-17, &c. or 3. Sometimes roots are designed like powers with fractional indices. Thus the square root of 15 is 154, the cube root of 21 is 21, and the 4th root of 37 20 is 37—20, &c. 9th Power 1 512 | 19683 | 262144 | 1953125 | 10077696 | 40353607 10th Power | 1| 1024 | 59049 | 1048576 | 9765625 | 60466176 | 282475249 | 1073741824 | 3486784401 134217728 387420489 SECTION LIII. THE EXTRACTION OF THE SQUARE ROOT. 1. What is the square root of 576 ? OPERATION. To illustrate this question, I will suppose, 576(24 Ans. that I have 576 tile, each of which is one foot square; I wish to know the side of a square room, whose floor they will pave or cover. 400 44)176 176 If we find a number multiplied by itself, that will produce 576, that number will be the side of a square room, which will require 576 tiles to cover its floor. We perceive, that our number (576) consists of three figures; therefore there will be two figures in its root; for the product of any two numbers can have at most, but just so many figures as there are in both factors; and at least, but one less. We will therefore, for convenience, divide our number (576) into two parts, called periods, writing a point over the right hand figure of each period; thus 576. We now find by the table of powers, that the greatest square number in the left hand period 5 (hundred) is 4 (hundred ;) and that its root is 2, which we write in the quotient. (See operation.) As this 2 is in the place of tens, its value must be 20, and its square 400. Let this be represented by a square, whose sides measure 20 feet each, and whose contents will therefore be 400 square feet. See figure D. We now subtract 400 from 576 and there remain 176 square feet to be arranged on two sides of the figure D, in order that its form may remain square. We therefore double the root 20, one of the sides, and it gives the length of the two sides to be enlarged: viz. 40. We then inquire how many times 40 as a divisor Fig. 1. 20 D 20 20 20 400 20 20 20 20 is contained in the dividend (except the right hand figure) and find it to be 4 times: this we write in the root and also in the divisor. This 4 is the breadth of the addition to our square. (See figure 2.) And this breadth, multiplied by the length of the two additions, (40,) gives the contents of the two figures, E and F, 160 square feet, which is 80 feet for each. There now remains the figure G, to complete the square, each side of which is 4 feet; it being equal to the breadth of the additions E and F. Therefore, if we square 4 we have the contents of the last addition G=16. It is on account of this last addition, that the last figure of the root is placed in the divisor. If we now multiply the divisor, 44, by the last figure in the root, (4,) the product will be 176, which is equal to the remaining feet after we had formed our first square, and equal to the additions E, F, and G, in figure 2. We therefore perceive, that figure 2 may represent a floor 24 feet square, containing 576 square feet. From the above we infer the following RULE. D contains 400 square feet. 80 do. do. E F G 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, and subtract it therefrom, and to the remainder bring down the next period for a dividend. 3. Place the 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. 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 in the root as last directed, and continue the operation in the same manner, till you have brought down all the periods. NOTE 1st. 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. NOTE 2d. 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 numbers are exhausted, the operation may be continued at pleasure by annexing ciphers. 2. What is the square root of 278784 ? 278784(528 Answer. 25 102)287 1048)8384 3. What is the square root of 776161 ? 8. Extract the square root of 234.09. Ans. 4.16. Ans. 19.3132079+. Ans. 2.98831055+. Ans. 1.4. Ans. 1.77482393+. 15. What is the square root of 572199960721 ? Ans. 756439. If it be required to extract the square root of a vulgar fraction, reduce the fraction to its lowest terms, then extract the square root of the numerator for a new numerator, and of the denominator for a new denominator; or reduce the vulgar fraction to a decimal, and extract its root. 16. What is the square root of? 17. What is the square root of 29? 18. What is the square root of 42 ? 8208 Ans. Ans. Ans. 6. |