of the root is contained wholly in the first two periods of the power; and so on. NOTE. The periods and figures of the root are counted from the left hand. The combinations in the formation of a square may be shown as follows: If we take any number consisting of two figures, as 43, and decompose it into two parts, 40+ 3, then the square of the number may be formed by multiplying both parts by each part separately: thus, 40+ 3 40+ 3 120+9 1600 + 120 432 = 1600 +240 + 9 Of these combinations, we observe that the first, 1600, is the square of 40 the second, 240, is twice 40 multiplied by 3; and the third, 9, is the square of 3. Hence, III. The square of a number composed of tens and units is equal to the square of the tens, plus twice the tens multiplied by the units, plus the square of the units. By observing the manner in which the square is formed, we perceive that the unit figure must always be contained as a factor in both the second and third parts; these parts taken together, may therefore be factored, thus, 240 +9 (803) × 3. Hence, IV. If the square of the tens be subtracted from the entire square, the remainder will be equal to twice the tens plus the units multiplied by the units. 1. What is the square root of 5405778576? its root, 7, as the first figure of the required root, and regard it as tens of the next inferior order, (II). We now subtract 49, the square of the first figure of the root, from the first period, 54, and bringing down the next period, obtain 505 for a remainder. And since the square of the first two figures of the root is contained wholly in the first two periods of the power, (II), the remainder, 505, must contain at least twice the first figure (tens) plus the second figure (units), multiplied by the second figure, (IV). Now if we could divide this remainder by twice the first figure plus the second, which is one of the factors, the quotient would be the second figure, or the other factor. But since we have not yet obtained the second figure, the complete divisor can not now be employed; and we therefore write twice the first figure, or 14, at the left of 505 for a trial divisor, regarding it as tens. Dividing the dividend, exclusive of the right hand figure, by 14, we obtain 3 for the second, or trial figure of the root, which we annex to the trial divisor, 14, making 143, the complete divisor. Multiplying the complete divisor by the trial figure 3, and subtracting the product from the dividend, we have 76 for a remainder. We have now taken the square of the first two figures of the root from the first two periods; and since the square of the first three figures of the root is contained wholly in the first three periods, (II), we bring down the third period, 77, to the remainder, 76, and obtain for a new dividend 7677, which must contain at least twice the two figures already found plus the third, multiplied by the third, (IV). Therefore to obtain the third figure, we must take for a new trial divisor twice the two figures, 73, considered as tens of the next inferior order, which we obtain in the operation by doubling the last figure of the last complete divisor, 143, making 146. Dividing, we obtain 5 for the next figure of the root; then regarding 735 as tens of the next inferior order, we proceed as in the former steps, and thus continue till the entire root, 73524, is obtained. 657. From these principles and illustrations we derive the following RULE. I. Point off the given number into periods af two figures each, counting from units' place toward the left and right. II. Find the greatest square number in the left hand period, and write its root for the first figure in the root; subtract the square number from the left hand period, and to the remainder bring down the next period for a dividend. III. At the left of the dividend write twice the first figure of the root, for a trial divisor; divide the dividend, exclusive of its right hand figure, by the trial divisor, and write the quotient for a trial figure in the root. IV. Annex the trial figure of the root to the trial divisor for a complete divisor; multiply the complete divisor by the trial figure in the root, subtract the product from the dividend, and to the remainder bring down the next period for a new dividend. V. Multiply the last figure of the last complete divisor by 2 for a new trial divisor with which proceed as before. NOTES.-1. If at any time the product be greater than the dividend, diminish the trial figure of the reot, and correct the erroneous work. 2. If a cipher occur in the root, annex a cipher to the trial divisor, and another period to the dividend, and proceed as before. 3. If there is a remainder after all the periods have been brought down, annex periods of ciphers, and continue the root to as many decimal places as are required. 4. The decimal points in the work may be omitted, care being taken to point off in the root according to the number of decimal periods used. 5. The square root of a common fraction may be obtained by extracting the square roots of the numerator and denominator separately, provided the terms are perfect squares; otherwise, the fraction may first be reduced to a decimal. 6. Mixed numbers may be reduced to the decimal form before extracting the root; or, if the denominator of the fraction is a perfect square, to an improper fraction. 7. The pupil will acquire greater facility, and secure greater accuracy, by keeping units of like order under each other, and each divisor opposite the corresponding dividend, as shown in the operation. EXAMPLES FOR PRACTICE. Ans. 562. Ans. 12345. Of 597?' 1. What is the square root of 315844? Ans. 103.9. Ans. 7.621. Ans. .00563. Ans. 62.11342. Ans. .745355+. Ans. 13 16' Ans. 202500. CONTRACTED METHOD. 658. 1. Find the square root of 8, correct to 6 decimal places. OPERATION. 12.828427+, Ans. 4 48 400 384 562 1600 1124 5648 47600 45184 5656 2416* 2262 566 154 ANALYSIS. Extracting the square root in the usual way until we have obtained the 4 places, 2.828, the corresponding remainder is 2416, and the next trial divisor, with the cipher omitted, is 5656. We now omit to bring down a period of ciphers to the remainder, thus contracting the dividend 2 places; and we contract the divisor an equal number of places by omitting to annex the trial figure of the root, and regarding the right hand figure, 6, as a rejected or redundant figure. We now divide as in contracted division of decimals, (226), bringing down each divisor in its place, with one redundant figure increased by 1 when the rejected figure is 5 or more, and carrying the tens from the redundant figure in multiplication. We observe that the entire root, 2.828427+, contains as many places as there are places in the periods used. Hence the following RULE. I If necessary, annex periods of ciphers to the given number, and assume as many figures as there are places required in the root; then proceed in the usual manner until all the assumed figures have been employed, omitting the remaining figures, if any. II. Form the next trial divisor as usual, but omit to annex to it the trial figure of the root, reject one figure from the right to form each subsequent divisor, and in multiplying regard the right hand figure of each contracted divisor as redundant. NOTES.-1. If the rejected figure is 5 or more, increase the next left hand figure by 1. 2. Always take full periods, both of decimals and integers. EXAMPLES FOR PRACTICE. 1. Find the square root of 32 correct to the seventh decimal Ans. 5.6568542+. place. 2. Find the square root of 12 correct to the seventh decimal place. Ans. 3.4641017+. 3. Find the square root of 3286.9835 correct to the fourth decimal place. Ans. 57.3322+. 4. Find the square root of .5 correct to the sixth decimal place. Ans. .745355+. 5. Find the square root of 6 correct to the sixth decimal place. Ans. 2.563479+. 6. Find the square root of 1.065 correct to the sixth decimal place. Ans. 1.156817+. 7. Find the value of 1.0125 correct to the fourth decimal place. Ans. 1.0188+. 8. Find the value of 1.023375% correct to the sixth decimal place. Ans. 1.01162+. CUBE ROOT. 659. The Cube Root of a number is one of the three equal factors that produce the number. Thus, the cube root of 343 is 7, since 7 x 7 x 7 = 343. To derive the method of extracting the cube root of a number, it is necessary to determine 1st. The relative number of places in a given number and its cube root. 2d. The relations of the figures of the root to the periods of the number. 3d. The law by which the parts of a number are combined in the formation of a cube; and 4th. The factors of these combinations. 660. The relative number of places in a given number and its cube, is shown in the following illustrations: |