THE SQUARE ROOT. THE square of a number is the product arising from that number multiplied into itself. Extraction of the square root is the finding of such a number. as, being multiplied by itself, will produce the number proposed. RULE. 1. Distinguish the given number into periods of two figures each, beginning at the units place, or decimal point; and when the decimal does not consist of an even number of figures, annex a cypher; and equal to the periods of whole numbers and decimals respectively will be the places of each in the root. 2. Deduct from the first period the greatest square it contains, setting the root thereof as a quotient figure, and doubling it for a divisor; and bring down the second period to the remainder, for a dividual. 3. Try how often the said divisor, with the resulting figure of this trial thereto annexed, are contained in the dividual, and set this resulting figure to both the divisor and root; then multiply and subtract as in division, and bring down the next period. 4. Double the ascertained root for a new divisor, and repeat the process to the end. PROOF. Square the root, adding in the remainder (if any) which will equal the number given. EXAMPLES. 1 What is the square root of 30138,696025 P 30138,696025(173,605 answer 1 27)201 189 343)1238 $466)20969 347205) 1736025 1736025 0 Note. When one more than half the figures of the root are found, the rest may be obtained by working as in contracted division of decimals. 2 Required the square root of 14876,2357 ? facit 2345 answer 8647 607,34092+ 57,19+ 3 Required the square-root of 5499025 P 1,50701+ 3,162277+ facit 12,649+per. 11 A certain number of men gave 30s 1d. for a charitable purpose; each man giving as many pence as there were men: query the number? answer 19 men. 12 If a circular pipe of 1,5 inches diameter fill a cistern in 5 hours; in what time would it be filled by one of 3,5 inches diameter ? answer 55min. 6sec. 13 If 484 trees be planted in a square orchard, how many must be in a row each way? answer 22 Note 1. The square of the longest side of a right angled triangle is equal to the sum of the squares of the other two sides; and consequently the difference of the square of the longest, and either of the other, is the square of the remaining side. 2. The square root of a vulgar fraction is found by reducing it to its lowest terms, and extracting the rot of the numerator, for a numerator, and of the denominator, for a denominator. Ifit be a surd, reduce it to its equivalent decimal, &e. 3. Amixt number may be reduced to an improper fraction, or a decimal, and the root thereof extracted as before. 14 The wall of a fortress is 17 feet high, which is surrounded by a moat 20 feet in breadth; query the length of a ladder to reach from the outside of the moat to the top of the wall? answer 26,2 feet. 15 A line of 36 yards long will exactly reach from the top of a fort to the opposite bank of a river, known to be 24 yards broad; the height of the wall is required? answer 26,83+yards. 16 Suppose a ladder, 60 feet long, be so planted as to reach a window 37 feet from the ground on one side of the street, and without moving it at the foot, will reach a window 23 feet high on the other side; what breadth was the street of? 3 answer 102,64 feet. 17 What is the square-root of 44 P 16 THE CUBE ROOT. HE Cube of a number is the product of that number multiplied into its square. Extraction of the cube root is the finding of such a number, as, being multiplied into its square, will produce the number proposed. RULE. First, Distinguish the proposed number into periods of three figures each, beginning at the units place, or decimal N 2 point point: and when the decimal does not consist of a complete period or periods, annex a cipher or ciphers to make it so; and the places of the root will be as many as the periods of the given cube in whole numbers and decimals respectively. Secondly, Find the greatest root of the left hand period, which place to the right of the given number, and subtract the cube thereof from said period; and to the remainder bring down the next period for a dividual. Thirdly, Take the triple square of the ascertained root for a defective divisor. Fourthly, Reserve mentally the units and tens of the dividual, and try how often the defective divisor is contained in the rest; place the result of this trial to the root, and its square to the right of said divisor, supplying the place of tens with a cipher, if the square be less than 10. Fifthly, Complete the divisor, by adding thereto the product of the last figure of the root by the rest, and by 30. Sixthly, Multiply, subtract, and bring down the next period for a dividual, for which find a divisor as before; and so proceed with every period. Note. Defective divisors, after the first, may be more concisely found by addition, thus: To the last complete divisor, add the number which completed it, with twice the square of the last figure in the root; the sum will be the next defective divisor. EXAMPLES. 1 What is the cube root of 444194,947 ? = Defec. div. & sqr. of 6 14736)101194 +1260 complete divisor 159976) 95975 Defec. div. & sqr. of S 1732809)5218947 +6840 complete divisor 1739649)5218947 0 answer 325 439 638 2805 8765 2,35 ,124 25,16+ 5,555+ 2 What is the cube-root of 34328125 ? 3 What is the cube-root of 84604519 ? 4 What is the cube-root of 259694072? 5 What is the cube-root of 22069810125 ? 6 What is the cube-root of 673373097125 ? 7 What is the cube-root of 12,977875 ? 8 What is the cube-root of ,001906624 ? 9 What is the cube-root of 15926,972504 ? 10 What is the cube-root of 171,46677406 P 11 What is the difference between half a solid foot, and a solid half foot? answer 3 half feet. 12 In a cubical foot, how many cubes of 6 inches, and how many of three, are contained thereein ? answer 8 of 6in. and 64 of sin. 13 The content of an oblong cellar is 1958,125 cubic feet; required the side of a cubical cellar that shall contain just as much? answer 12,5 feet. 14 A stone of a cubic form contains 474552 solid inches; what is the superficial content of one of its sides ? answer 6084 inches. 15 A merchant laid out 691 4s. in cloths, but forgot the number of pieces purchased, also how many yards were in each piece, and what they cost him per yard; but remembers, that they cost him as many shillings per yard as there were yards in each piece, and that there was just as many pieces: query the number purchased ? answer 24 Note 1. The cube root of a vulgar fraction is found by reducing it to its lowest terms and extracting the root of the numerator for a numerator, and of the denominator for a denominator. If it be a surd, extract the root of its equivalent decimal. 2. A mixt number may be reduced to an improper fraction, or a decimal, and the root thereof extracted. T188 3000 16 What is the cube-root of 352 ? 125 125 answer ,763 ,949+ 2,3908+ 31 1,966+ 2,092+ GENERAL |