any) both in the quotient, and on the right of the di Multiply the divisor (thus augmented) by the figure last the quotient, and set the product under the dividend. - Subtract, and bring down the next period to the reer for a dividend; and to the left of this bring down the visor with its right hand figure doubled, for a divisor. I. Find how often the divisor is contained in the dividend, ng the units as before; put the number denoting how in the quotient, and also on the right of the divisor. ply, subtract, bring down the next period, and also the r with its right hand figure doubled, &c. as before, and ed in this manner till the work is finished. If there is a remainder, periods of ciphers may be succesbrought down, and the work continued as before, observat the quotient figures which arise will be decimals; and re be an odd decimal figure in the given number, a cipher be subjoined, to make the right hand period complete. for every period of superfluous ciphers, either on the left or of the given number, a cipher must be placed in the quoThe operations may be proved by involving the root to quare, (Art. 265.) and adding in the remainder, if any. EXAMPLES. Extract the square root of 54756. OPERATION. 54756(234 =root 4 147 129 1856 1856 Proof. 234 Explanation. first place a point over the units, then over the hundreds, then over the ten thousands; 5 being the first period, I find from the table the greatest square 4, contained in it; this 4 I place under the 5, and its root 2 in the quotient, and having subtracted, I bring down to the remainder I the next period 47, making 147 for the dividend; I double the quotient figure 2, and place the double, viz. 4, for a divisor, to the left. Omitting the units 7, I ask how often 4 is contained in 14, and find it goes 3 times; this 3 I put both in the quotient and divisor, making the latter 43; this I multiply by the quotient figure 3, and subtract the product 129 from the dividend. To the remainder 18 I bring down the next period 56, making the new dividend 1856; to the left of this I bring down the divisor with its last figure 3 doubled, making 46: I then ask how often 46 goes in 185, (omitting the 6,) it goes 4 times, I therefore put the 4 both in the quotient and on the right he divisor, and multiply as before: there being neither a remainder, nor more figures to bring down, the operation is finished. root 234 = root 936 702 468 54756 square. 2. Extract the square root of .000064807. OPERATION". .0000648070).008050279 root. 64 1605) 8070 8025 161002) 450000 322004 1610047) 12799600 11270329 16100549) 152927100 144904941 Remainder 8022159 Explanation. Here, in order to complete the right hand period, I subjoin a cipher; and there being 2 periods of ciphers to the left, I prefix a cipher for each period to the root. In the second step, having brought down the 80, I find that 16 will not go in 8; I therefore put a cipher both in the quotient and divisor, and then bring down the next period 70; and the like in the next step. I bring down a period of ciphers both there and in each following step. 3. Extract the square root of 95.801234. 95.801234(9.78781 root. "These operations may be proved three ways. First, by involving the root to the square as in ex. 1. and adding the remainder to the square: the result, if the work be right, will equal the given number. Secondly, by casting out the nines: thus, cast out the nines from the root, and multiply the excess into itself; cast the nines out of the product, reserving the excess; cast the nines out of the remainder, subtract the excess from the dividend, and cast the nines out of what remains if this excess equals the former, the work may (with the restriction mentioned in the note on Art. 41.) be presumed to be right. Thirdly, by addi. tion, similar to the proof of long division, Art. 41. Root 987. Root 1023. 7. Extract the square root of 974169. 8. What is the square root of 1046529 ? 9. Extract the square root of 867.8916. 10. Required the square root of 32.72869681. Root 29.46. Root 5.7209. 11. Find the square root of 70. Root 8.3666, &c. 12. What is the square root of .000294? Root .0171464, &c. 13. What is the square root of 989? Root 31.44837, &c. 14. Find the square root of 6.27. Root 2.503996805, &c. 15. Required .00015241578750190521. Root .0123456789. 274. To extract the square root of a vulgar fraction, both terms of which are exact squares. RULE. Extract the root of the numerator, and likewise of the denominator; these two roots will be respectively the terms of a new fraction, which will be the root required. 16. Extract the square root of 4 9 Explanation. OPERATION. 2 3 Here the root of 4 is 2, and the root of 2 the root required. 9 is 3, therefore will be the root of 3 275. To extract the square root of a vulgar fraction, the terms of which are not both squares. RULE. Reduce the given fraction to a decimal, (Art. 233.) and extract the root of this decimal for the answer. 18. What is the square root of First 14 1 ? = .5 by Art. 233. Then 5.70710678119, &c. (Art. 973.) the root required. 276. To extract the square root of a mixed number. RULE. Reduce the fraction to a decimal, (Art. 233.) to which prefix the whole number, extract the square root of the result by Art. 273. and it will be the root required. 21. To find the square root of 8 3 4 Then (Art. 273.) √/8.75 = 2.95803989, the root required. 22. What is the square root of 13 4동 14 30 10 Thus (Art. 178.) = .344827586206, &c. Therefore 45 87 29 344827586206, &c. = .5872202, &c. (Art. 277. Sometimes it happens, that the given mixed number being reduced to its equivalent improper fraction, both the terms will be rational. In this case it will be best to extract the roots of the numerator and denominator separately, and they will form an improper fraction, which must be reduced to its proper terms, (Art. 173.) 26. Extract the square root of 24. 9 Thus (Art. 172.) 24 = 2. Then (Art. 274.) √ 4 =14 (Art. 173.), the root required. 9 278. PROMISCUOUS. EXAMPLES FOR PRACTICE. 1. The side of a square kitchen garden is 63 yards; how many square yards does the garden contain? Ans. 3969 square yards. 2. An army consisting of 87616 men, is to be arranged in the form of a square; how many will a side contain? Ans. 296 men. 3. What will the paling of a square garden consisting of 2209 square yards cost, at 3s. 6d. per yard? Ans. 321. 18s. 4. Required the length of a ladder, the foot of which being pitched 12 feet from the wall, its top will reach a window 18 feet from the ground *? Ans. 21 feet, 7.59969, &c. inches. 5. A rope 120 fathoms long is extended from the top of a cliff, to a boat moored at SO fathoms from its base; required the perpendicular height of the cliff? Ans. 89.4427, &c. fathoms. 6. Required a mean proportional between 36 and 2401? Ans. 294. 7. A rectangular field has its sides equal to 210 and 300 yards respectively; what length must the side of a square be to contain an equal area? Ans. 250.9979, &c. yards. 8. The diagonal (or straight line joining the opposite corners) of a chess-board, measures 30 inches; required the length of the side? Ans. 21.213203, &c. inches. 9. A wall is supported 13 feet from the ground by a shoar 16 feet long; how far is the foot of the shoar distant from the base of the building? Ans. 9 feet, 3.9285, &c. inches. 10. A gentleman has a table 5 feet wide, and 14 feet long, and wants three others to be made, each square, and all together of equal dimensions with the former; required the side of each? Ans. 4 feet, 9.9654, &c. inches. EXTRACTION OF THE CUBE-ROOT. 279. The following table contains the first nine whole num The length of the ladder, the perpendicular distance of its top from the ground, and the distance of its foot from the wall, together form a right angled triangle; and it is demonstrated in the 47th proposition of the first book of Euclid's Elements, that the square of the longest side of such triangle is equal to the sum of the squares of the two remaining sides; wherefore in the present instance, 12+18)2 = 21.633, &c. feet, the length required. y From the foregoing note it follows, that the difference of the squares of the longest and of either of the remaining sides, is equal to the square of the other 2 ---side; whence 120 -80289.44, &c. fathoms. 2 The square root of the product of any two numbers is a mean proportional between them. |