8. What is the 6th power of 13? Ans. 1613144 9. What is the 4th power of .045 ? Ans. .000004100625. 10. What is the O power of 1728 ? Ans. 1. EVOLUTION, OR THE EXTRACTION OF ROOTS. EVOLUTION is the reverse of Involution, and teaches to find the roots of any given powers. The root is a number whose continual multiplication into itself produces the power, which is denominated the 2d, 3d, 4th, &c., power, according to the number of times which the root is multiplied into itself. Thus, 4 is the square root of 16, because 4x4 = 16; and 3 is the cube root of 27, because 3 X3 X 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 be. fore 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 tor- - 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. Sometimes roots are designated like powers with fractional indices. Thus the square root of 15 is 153, the cube root of 21 is 214, and the 4th root of 37 – 20 is 37 — 204, &c. It sometimes will happen that one root is involved in another, thus : o 125 -5 + 19 +6, or 161 — 147. 3 ♡ V 178 +7 933 – 8+ 384 – 5 – 387 + 16. 1 4 9 16 25 36 49 64 81 3d Power 11 8 27 64 125 216 343 512 729 4th Power | 1 16 81 256 625 1296 2401 4096 6561 512 19683 262144 1953125 10077696 40353607 1024 59049 | 1048576 | 9765625 60466176 | 282475249 | 2d Power 5th Power 8th Power 9th Power 10th Power 1 1073741824 | 3486784401 SECTION LXI. OPERATION. EXTRACTION OF THE SQUARE ROOT. 1. Let it be required to find what number multiplied into itself will produce 1296. In contemplating this 1296(30+6= 36 Ans. problem, we perceive 900 that the root or number 60+6=66)396 sought must consist of 396 two figures, since the product of any two numbers can have at most but as many figures as there are in both factors, and at least but one less. We perceive, also, that the first figure of the root multiplied by itself must give a number not exceeding 12, and as 12 is not the second power of any number, and the second power of 4 is more than 12, we take 3 for the first figure of the root, which multiplied into itself gives 9. Now, the second power of 3, considered as occupying the place of tens, is 900, of which we have the root 30. Taking 900 from 1296, we have a remainder, 396; and having found the root of 900, we are now to seek a number, which, being added to this root (30) and multiplied into itself once, and into 30 twice,* will produce 396. This number is found by dividing 396 by twice 30 plus the number sought. Q. E. D. Note. — Owing to the fact that the number of figures in the product of any two numbers is always limited as above stated, we ascertain the number of figures in the root of any given second power by putting a dot over the place of units, then over the place of hundreds, and so on. The number of dots gives the number of figures in the root. Thus the square root of 133225 consists of three figures. 2. What is the square root of 576 ? To illustrate this question in a different 576(24 Ans. way from the first, we will suppose that we 400 have 576 tiles, each of which is one foot square, and we wish to know the side of a 44) 176 square room whose floor they will pave or 176 OPERATION. cover. * By adding 6 to 30 and multiplying the sum (36) into itself, we can easily see that we multiply 6 by itself once, and 30' by 6 twice, since 30 is contained in both factors, and in the operation is multiplied by the 6 in each. 20 20 20 If we find a number which multiplied into itself will produce 576, that number will give the side of the room required. We perceive that as our number (576) consists of three figures, there will be two figures in its root, since the square of no number expressed by a single figure can be so large as 576 ; and, if the root were supposed to have more than two figures, its square would exceed 576. Dividing the number into periods thus, 576, we now find by trial, or by the table of powers, that the greatest square number or second power 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 or second power 400. Let this be represented by a square whose sides measure 20 feet each, and Fig. 1. whose contents will therefore be 400 square feet. See figure D. We now subtract 400 from 576 and there remain 176 D square feet to be arranged on two sides of 20 the figure D, in order that its form may 20 remain square. We therefore double the 400 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 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 E the contents of the two figures E and F, 160 square feet, which is 80 feet for each. D F There now remains the space G, to 20 20 20 complete the square, each side of which is 4 feet; it being equal to the 400 80 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 divi. sor, 44, by the last figure in the root (4), the product will be 20 Fig. 2. 20 4 GA 4 4 16 80 20 20 66 66 176, which is equal to the remain- D contains 400 square feet. ing feet after we had formed our E 80 first square, and equal to the addi. F 80 tions E, F, and G, in figure 2. G 16 We therefore perceive that fig. Proof, 576 ure 2 may represent a floor 24 feet square, containing 576 square 24 x 24=576 feet. or RULE. — 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, placing 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, subtracting it therefrom; and to the remainder bring down the next period for a dividend, always, however, omitting the right-hand figure of this dividend in dividing: 3. Place the double of the root already found on the left hand of the dividend for a divisor. 4. Find how often the divisor is contained in the dividend (omitting the right-hand figure), placing 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 the last divisor for a new one, doubling the right-hand fig; ure 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 1.- If, when the given power is pointed off as the case requires, the left-hand period should be deficient, it must nevertheless stand as the first period. Note 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 numbers are exhausted, the operation may be continued at pleasure by annexing ciphers. * One or two units are generally to be allowed on account of other deficiencies in enlarging the square. |