Ex. 7. Required the square root of x+2x+y+ +ys. Ans. x+y. Ex. 8. Required the square root of x3 —2x1y*+y3. Ans. x-y. 4 Ex. 9. Required the cube root of a3-6a2x+12ax2 -8x3. Ex. 10. Required the sixth root of x-6x+15xa -20x+15x2-6x+1. Ans.x-1. 8 Ex. 11. Required the fifth root of x10+15x3y2+ 96x*y*+270x1y° +405x2y3+243y1o. Ans. x2+3y2. Ex. 12. Required the square root of x2+2xy +y2 +6xz+6yz+9z2. Ans. x+y+32. § III. INVESTIGATION OF THE RULES FOR THE EX TRACTION OF THE SQUARE AND CUBE ROOTS OF NUMBERS. 304. It has been observed, (Art. 104), that, a denoting the tens of a number, and b the units, the formula a2+2ab+b2 would represent the square any number consisting of two figures or digits; as, for example, if we had to square 25; put a=2and b= 5, and we shall find a2=400 2ab=200 b2 = 25 (a+b)2=(25)2=625. 305. Before we proceed to the investigation of these Rules, it will be necessary to explain the nature of the common arithmetical notation. It is very well known that the value of the figures in the common arithmetical scale increases in a tenfold proportion from the right to the left; a number, therefore, may be expressed by the addition of the units, tens, hundreds, &c. of which it consists; thus the number 4371 may be expressed in the following manner, viz. 4000+300+70+1, or by 4x1000+3x100+7x10 +1; also, in decimal arithmetic, each figure is supposed to be multiplied by that power of 10, positive or negative, which is expressed by its distance from the figure before the point: tus, 672.53=6 × 102+7x101 +2×10° +5 X 10-1+3 10-26 x 100+7x10+2 50 3 5 3 x1+-+ 10 100 53 672+ + -672- Hence, if 100 100 100 the digits of a number be represented by a, b, c, d, e, &c. beginning from the left-hand; then, A number of figures may be expressed by 10a+b. 3 figures by 100a10b+c. 4 figures by 1000a +100b+10c+d. &c. &c. &c. By the digits of a number are meant the figures which compose it, considered independently of the value which they possess in the arithmetical scale. Thus the digits of the number 537 are simply the numbers 5, 3 and 7; whereas the 5, considered with respect to its place, in the numeration scale, means 500, and the 3 means 30. 306. Let a number of three figures, (viz. 100a +10b+c) be squared, and its root extracted according to the rule in (Art. 299), and the operation stands thus; I. 10000a2+2000ab+10063+200ac+20bc+c2 10000a2 (100a +10b+c 200a10b)2000ab+100b2 2000ab+10062 200a +20b+c)200ac+20bc+c2 40000+12000+900+400+60+1(200+30+1 40000 400+30)12000+900 400+60+1)400+60+1 400+60+1 III. But it is evident that this operation would not be affected by collecting the several numbers which stand in the same line into one sum, and leaving out the ciphers which are to be subtracted in the operation.. Let this be done; and let two figures be brought down at a time, after the square of the first figure in the root has been subtracted; then the operation may be exhibited in the manner annexed; from which it appears, that the square root of 53361 is 231. 307. To explain the division of the given number into periods consisting of two figures each, by placing a dot over every second figure beginning with the units, as exhibited in the foregoing operation. It must be observed, that, since the square root of 100 is 10; of 10000 is 100; of 1000000 is 1000 ; &c. &c. it follows, that the square of a number less than 100 must consist of one figure; of a number between 100 and 10000, of two figures; of a number between 10000 and 1000000, of three figures; &c. &c., and consequently the number of these dots will show the number of figures contained in the square root of the given number. From hence it follows, that the first figure of the root will be the greatest square root contained in the first of those periods reckoning from the left. Thus, in the case of 53361 (whose square root is a number consisting of three figures); since the square of the figure standing in the hundred's place cannot be found either in the last period (61), or in the last but one (33), it must be found in the first period (5); consequently the first figure of the root will be the square root of the greatest square number contained in 5; and this number is 4, the first figure of the root will be 2. The remainder of the operation will be readily understood by comparing the steps of it with the several steps of the process for finding the square root of (a+b+c)2 (Art. 299); for, having subtracted 4 from (5), there remains 1; bring down the next two figures (33), and the dividend is 133; double the first figure of the root (2), and place the result 4 in the divisor; 4 is contained in 13 three times; 3 is therefore the second figure of the root; place this both in the divisor and quotient, and the former is 43; multiply by 3, and subtract 129, the remainder is 4 to which bring down the next two figures (61), which gives 461 for a dividend. Lastly, double the last figure of the former divisor, and it becomes. 46 place this in the next divisor, and since 4 is contained ; in 4 once, 1 is the third figure of the root; place 1 therefore both in the divisor and quotient; multiply and subtract as before, and nothing remains. 308. The method of extracting the cube root of numbers may be understood by comparing the process for extracting the cube root of (a+b+c), (Art. 300), with the following operations, in which is deduced the cube root of the number 13997521. 13997521(200+40+1 a3=(200)3=8000000 1st remainder 5997521 3a2=3×(200)2= divisor, .... 3a3b=3(200)2 × 40=4800000 = b3=40×40 X 40= 64000 5824000 2nd remainder 173521 3(a+b)'c=(200+40)2 x1=172800 3(a+b)c2=3(200+40)×1= c3=1X1X1= 720 1 173521 3d remainder 000000 Omitting the superfluous ciphers, and bringing down three figures at a time, the operation will stand thus ;; |