260. The powers of the nine digits, from the first to the sixth inclusive, are exhibited in the following 251 TABLE. Roots, or 1st powers, 111 21 31 41 51 61 71 8 91 Squares, or 2d powers, 111 41 9! 16) 36 49 64 81! Cubes, or 3d powers, 111 8127 64 125 216 343 512 729 Biquadrates, or 4th p. 11|16|81| 256| 625 1296 2401 4096 6561 Sursolids, or 5th power 11/32/243|1024| 3125) 7776] 168071 32768| 59049 Square cubes, or 6th p. 111641729|4096|15625146656|117649|2621441531441 2. Evolution. ANALYSIS. 261. The method of ascertaining, or extracting the roots of numbers, or powers, is called Evolution. The root of a number, or power, is a number, which, multiplied by itself continually, a certain number of times, will produce that power, and is named from the denomination of the power, as the square root, cube root, or 2d root, 3d root, &c. Thus 27 is the cube or 3d power of 3, aud hence 3 is called the cube, or 3d, root of 27. 262. The square root of a quantity may be denoted by this character called the radical sign, placed before it, and the other roots by the same sign, with the index of the root placed over it, or by fractional indices placed on the right hand. Thus, V9, or 9, denotes the square root of 9, 327, or 27t, denotes the cube root of 27, and 16, denotes the 4th root of 16. The latter method of denoting roots is preferable, inasmuch as by it we are able to denote roots and powers at the same time. Thus, 88 signifies that 8 is raised to the second power, and the cube root of that power extracted, or that the cube root of 8 is extracted, and this root raised to the second power; that is, the numerator of the index denotes the power, and the denominator the root of the number over which it stands. or 167, 263. Although every number must have a root, the roots of but very few numbers can be fully expressed by figures. We can, however, by the help of decimals approximate the roots of all sufficiently near for all prac. tical purposes. Such roots as cannot be fully expressed by figures are denominated surds, or irrational numbers. 264. The least possible root, which is a whole number, is 1. The square of 1 is (1x4) 1, which has one figure less than the number employed as factors; the cube of 1 is (1x1x1=),1, two figures less than the number employed as factors, and so on. The least root consisting of two figures is 10, whose square is (10X10_) 100, which has one figure less than the number of figures in the factors, and whose cube is (10x10x 10_) 1000, two figures less than the number in the factors; and the same may be shown of the least roots consisting of 3, 4, &c. figures. Again, the greatest root consisting of only one figure, is 9, whose square is (9X93) 81, which has just the number of figures in the factors, and whose cube is 1989 X9=) 729, just equal to the number of figures in the factors; and the greatest root consisting of two figures, is 99, whose square is (99X993) 9801, &c., and the same may be shown of the greatest roots consisting of 3, 4, &c. figures. Hence it appears that the number of figures in the contínued product of any number of factors cannot exceed the number of figures in those factors ; nor falt short of the number of figures in the factors by the number of fuctors, wanting one. From this, it is clear that a square number, or the second power, can have but twice as many figures as its root, and only one less than twice as many; and that the third power can have only three times as many figures as its root, and only two less than three times as many, and so ou for the higher powers. Therefore, 265. To discover the number of figures of which any root will consist. RULE.-Beginning at the right hand, distinguish the given number into portions, or periods, by dots, each portion consisting of as many figures as are denoted by the index of the root; by the number of dots will be shown the number of figures of which the root will consist. EXAMPLES. 1. How many figures in the 2. How many figures in the square, cube, and biquadrate square and cube root of 68101 root of 348753421 ? 2.1416 ? 34875342 i square root 5. 681012. 1416 square 5. 34875342 i cube root 3. 681012141600 cube 4. 34875242 i biquadrate 3. In distinguishing decimals, begin at the separatrix and proceed towards the right hand, and if the last period is incomplete, complete it by annexing the requisite number of ciphers. EXTRACTION OF THE SQUARE ROOT. ANALYSIS. 266. To extract the square root of a given number is to find a number, which, multiplied by itself, will produce the given number, or it is to fand the length of the side of a square of which the given number expressen 10 the area. 1. If 529 feet of boards be laid down in a square form, what will be the length of the sides of the square ? or, in other words, what is the square root of 529 ? From what was shown (264), we know the root must consist of two fig. ures, in as much as 529 consists of two periods. Now to understand the method of ascertaining these two figures, it may be well to consider how the square of a root consisting of two figures is formed. For this pur pose we will take the number 23, and 23 square it. By this operation, it appears 23 that the square of a number consisting of tens and units is made up of the 9 square of units. square of the units, plus twice the pro60 twice the product of } duct of the tens, by the units, plus the square of the tens. See this exhibited 400 square of the tens. in figure F. As 10x10=100, the square of the tens can never make a part of the 529 square of 23. two right hand figures of the whole square. Hence the square of the tens is always contained in the second peri5 29 (20 od, or in the 5 of the present example. 4 00 The greatest square in 5 is 4, and its root 2; hence, we conclude, that the 1 29 tens in the root are 2–20, and 20 x 20 400. But as the square of the tens can never contain significant figures below hundreds, we need only write the square of the figure denoting tens under the second period. From what precedes it appears that 400 of the 529 feet of boards are now disposed of in a square form, E measuring 20 feet on each 20 ft side, and that 129 feet are to be added to this square in such manner as not to alter its form; and in order to do this, the additions must be E made upon two sides of the square, E_20+ 20–40 feet. Now if 129, the number of feet to 20 be added, be divided by 40, the length of the i 20 additions, or, dropping the cipher and 9, if 12 be divided by 4, the quotient will be the width of 400 ft. the additions; and as 4 in 12 is had 3 times, we conclude the addition will be 3 feet wide, and 40X3=120 feet, the quantity added upon the two sides. But since these additions are no longer than the sides of the square, E, there must be a deficiency at the corner, as exhibited in F, whose sides 20 ft. 3 ft. are equal to the width of the additions, 20X360. or 3 feet, and 3X3=9 feel, required 16 fill out the corner, so as to complete the square. The whole operation may be arranged as on the next page, where it will be seen, that we first find the root of the greatest square in the left hand period, place it in the form of a quotient, 20x20=400 subtract the square from the period and to the remainder bring down the next period, which we divide, omitting the right hand figure, by double the root, and place the quotient for the second 23 ft. figure of the root; and the square of this 20 ft. 19 23 ft. 20 ft. 20x3=60. 529 ( 23 figure being necessary to preserve the form of the square, by filling the corner, we place it at the right of the divisor, in 43 ] 129 place of the cipher, which is always un129 derstood there, and then multiply the whole divisor by the last figure of the 23 X23529 proof. root. As we may conceive every root to be made up of tens and units, the above reasoning may be applied to any number whatever, and may be given in the following general RULE. 267. Distinguish the given numbers into periods ; find the root of the greatest square number in the left hand period, and place the root in the manner of a quotient in division, and this will be the highest figure in the root required. Subtract the square of the root already found from the left hand period, and to the remainder bring down the next period for a dividend, Double the root already found for a divisor; seek how many times the divisor is contained in the dividend (excepting the right hand figure), and place the result for the next figure in the root, and also on the right of the divisor. Multiply the divisor by the figure in the root last found; subtract the product from the dividend, and to the remainder bring down the next period for a new dividend. Double the root now found for a divisor, and proceed, as before, to find the next figure of the root, and so on, till all the periods are brought down. Ans. . QUESTIONS FOR PRACTICE. 1. What is the square root 6. What is the square root of 529? of Rz? 2. What is the square root Ans. .64549+. of 2? Ans. 1.4142+. Reduce to a decimal and The decimals are found by an- then extract the root (130). nexing pairs of ciphers continually to the remainder for a new dividend. 7. What is the square root In this way a surd root may be obtained to any assigned degree of of si? 3. What is the square root 8. What is the square root of 14? of 182.25 ? Ans. 13.5. 4. What is the square root 9. An army of 567009 men of .0003272481 ? are drawn up in a solid body, Ans. .01809. in form of a square; what is Hence the root of a decimal is the number of men in rank greater than its powers. and file ? Ans. 753. 5. What is the square root of 5499025? Ans, 2345. 10. What is the length of exactness. Ans. If the side of a square, which ameter of a circle 4 times as shall contain an aere, or 160 large ? Ans. 24. rods ? Ans. 12.649+ rods. Circles are to one another as the 11. The area of a circle is squares of their diameter ; therefore 234.09 rods; what is the length square the given diameters, multiply of the side of a square of or divide it by the given proportion, equal area? as the required diameters is to be greater or less than the given diamAns. 15.3 rods. eter, and the square root of the pro12. The area of a triangle duct, or quotient, will be the diamis 44944 feet; what is the eter required? length of the side of an equal 14. The diameter of a circle square ? Ans. 212 feet. is 121 feet; what is the diam13. The diameter of a circle i eter of a circle une half as is 12 inches; what is the di- | large ? Ans. 85.5+ feet. 268. Having two sides of a right angled triangle given to find the other side. RULE.-Square the two given sides, and if they are the two sides which include the right angle, that is, the two shortest sides, add them together, and the quare root of the sum will be the length of the longest side ; if not, the two shortest; subtract the square of the less from that of the greater, and the square root of the remainder will be the length of the side re, quired. (See demonstration, Part I. Art. 68.) QUESTIONS FOR PRACTICĘ. 1. In the right angled tri- If A B be 45 inches, and angle, A B C, the side A C is A C 36 inches, what is the 36 inches, and the side B C, length of B C? 27 inches; what is the length A B2=45X45–2025 of the side A B? A C2336X36–1296 |