SQUARE ROOT. ART. 232. The square root of any number, which is not a surd, may be determined by Resolving the number into its prime factors-the continued product of every other one of these different factors will be the root required. 1. What is the square root of 5184 ? OPERATION. 5184 = 2 × 2* × 2 × 2* × 2 × 2* × 3 × 3* × 3 × 3* EXPLANATION. Every other one of the different prime fac tors of 5184 is marked by *; the product of which is 2 × 2 × 2 × 3 × 3 = 72, the square root of 5184. 2. What is the square root of 900? ART. 233. The square root of any quantity which is not a surd, and is expressed by not more than four figures, can be ascertained by inspection. First, square the nine digits respectively, and observe the terminating figure of each square number. The terminating figures that are alike are linked together. We observe that all square numbers end in 1, 4, 9, 6, or 5; also, if the number ends in 9, the figure in the root occupying the unit's place must be either 3 or 7; if in 4, the figure in the root occupying the unit's place must be either 2 or 8, &c. The figures occupying the hundreds, or the hun dred's and thousand's place, will enable us to determine the figure of the root occupying the ten's place; and by the excess of the given quantity above the square of the ten's figure, we are enabled to tell which of the two figures that will produce the terminating figure of the quantity, is the root. 1. What is the square root of 5184 ? REMARF In accordance with what we have already learned, we know the figure in the root occupying unit's place must be 2 or 8; and the one occupying the ten's place must be 7, as its square, 49, is the largest square number, which is less than 51; and since the excess of the 51 above 49 is so small. we take the figure 2 for the unit's figure of the root. Hence, the square root of the above number is 72. Should the number have been 6084. then the excess of 60 above 49 would have been so great, we should have taken the 8 for the unit's figure of the root. Hence, we would have 78 for the square root of 6084. 2. What is the square root of 676? 3. What is the square root of 2209 ? 4. What is the square root of 1225? 5. What is the square root of 2916? 6. What is the square root of 3969 ? 7. What is the square root of 5041 ? 8. What is the square root of 7921? 9. What is the square root of 8464? 10. What is the square root of 9025? ART. 234. The square of 1, (the smallest digit,) is 1. The square of 9, (the largest digit,) is 81. Hence, the square of any digit is expressed by either one or two figures. The square of 10 (the smallest number denoted by two figures,) is 100. The square of 99, (the largest number denoted by two figures,) is 9801. Hence, the square of any number denoted by two figures, is expressed by either three or four figures; in the same manner it may be shown, that the square of any number denoted by three figures, will be expressed by either five or six figures, &c. Hence, the square of any number will contain twice as many figures as that number, or twice as many, less one. Therefore, to extract the square root, we first separate the num ber into periods of two figures each, commencing at the right ART. 235. As Evolution is the reverse of Involution, we will involve a few quantities by considering them decomposed into UNITS, TENS, HUNDREDS, &c., from which we will deduce a general rule for the extraction of the square root. The square of a binomial, that is, a quantity consisting of two terms, is equal to, The square of the first term, plus twice the first term into the second, plus the square of the second term. 1. What is the square of 35? 35 30 +5. Consider 30 the first term and 5 the second; then by the above rule we have, 35(30+5)2= 30+2x30x5+5=1225. : The evolution by multiplication is as follows: 302+2 × 30 × 5 +52 This involution may be geometrically illustrated thus:-Suppose the square ABCD, to be 30 inches each way; then its superficial contents is expressed by 30o. This square may be increased by the two rectangles ABFE and BCIH, each equal in length to the side of the square, and in width to BH, (or 5,) the quantity by which the square has been increased; hence, the area of E Α H B 302 D 30 x 5 each of these rectangles is expressed by 30 × 5; also the little square BHGF, whose side is BH, (or 5); hence, is area is 52. ART. 236. The square of any polynominal is equal to, The square of the first term, plus twice the first term into the second, plus the square of the second; plus twice the sum of the first two into the third, plus the square of the third; and so on. 1. What is the square of 452? 452 = 400 + 50 + 2. Consider 400 the first term, 50 the second term, and 2 the third term; then by the above theorem we have, (400+50+2)2=4002+2×400 X50+502+2x (40 +50) ×2+2o. By reversing the above process of involution, we obtain for extracting the square root, the following GENERAL RULE. Commencing at the right, separate the number into periods of two numbers each. Find the greatest square number in the first period on the left, and place its root at the right of the number, in the form of a quotient; also, on the left separating it from the number by a perpendicular line. Then subtract the square of this root from the period on the left; and to the remainder annex the second period; which will form the FIRST DIVIDEND. Double the root already found, (which is placed at the left of the number); to this product annex a cipher, and it will form the FIRST TRIAL DIVISOR. The number of times the trial divisor is contained in the first dividend, will be the next figure of the root, which must be added to the trial divisor, to form the TRUE DIVISOR. Multiply the true divisor by the figure of the root last obtained; subtract the product from the dividend, and to the remainder annex the next period for a NEW DIVIDEND. To the last divisor, add the last figure of the root found, this sum with a cipher annexed will be the next TRIAL DIVISOR. Then proceed as before, until all the periods have been brought down. NOTE. When any dividend is not so large as its trial divisor, place a cipher for the next figure of the root; also, place a cipher at the right of the divisor, and form a new dividend by annexing a new period. 1. What must be the length of the side of a square pond that shall contain 54756 square feet? It is evident that the side of the square must be more than 200 linear feet, since the square of 200 is less than 54756; also, that it must be less than 300 linear feet, since the square of 300 is greater than 54756. Therefore, 2 is the greatest number whose square is contain ed in 5, (the left hand period,) and is the first, or hundreds' figure of the root. Let CDEL be a square whose side is 200 linear feet. Then its area is 2002 40000 square feet, which being taken from the given number, leaves 14756 square feet, to be added to the square DI.. We first add the two rectangles CN and EM |