To extract the Square Root of a Mixed Number. RULE. Reduce the mixed,number to an improper fraction, and proceed as in the foregoing examples: Or, Reduce the fractional part to a decimal, annex it to the whole number, and extract the square root therefrom. EXAMPLES. 1. What is the square root of 3738? 2. What is the square root of 27 3. What is the square root of 85? 4. What is the square root of 84? APPLICATION. Ans. 64. Ans. 54. Ans. 9.27+ Ans. 2.9519+ 1. The square of a certain number is 105625: what is that number. Ans. 325. 2. A certain square pavement contains 20736 square stones, all of the same size, what number is contained in one of its sides? Ans. 144. 3. If 484 trees be planted at an equal distance from each other, so as to form a square orchard, how many will be in a row each way? Ans. 22. 4. A certain number of men gave 30s. 1d. for a charitable purpose; each man gave as many pence as there were men: how many men were there? Ans. 19. Note. 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 one. 5. The wall of a certain fortress is 17 feet high, which is surrounded by a ditch 20 feet in breadth; how long must a ladder be to reach from the outside of the ditch to the top of the wall? Ans. 26.24+-feet. 6. A certain castle which is 45 yards high, is surrounded by a ditch 60 yards broad; what length must a ladder be to reach from the outside of the ditch to the top of the castle? Ans. 75 yards. 7. A line 27 yards long, will exactly reach from the top of a fort to the opposite bank of a river, which is known to be 23 yards broad; what is the height of the fort? Ans. 14.142+ yards. 8. Suppose a ladder 40 feet long be so planted as to reach a window 33 feet from the ground, on one side of the street, and without moving it at the foot, will reach a window on the other side 21 feet high; what is the breadth of the street? Ans. 56.64+ feet. 9. Two ships depart from the same port; one of them sails due west 50 leagues, the other due south 84 leagues; how far are they asunder? Ans. 97.75+ Or, 973 + leagues. THE CUBE ROOT. The 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. 1. Separate the given number into periods of three figures each, beginning at the units place. 2. Find the greatest cube contained in the left hand period, and set its root on the right of the given number: subtract said cube from the left hand period, and to the remainder bring down the next period for a dividual. 3. Square the root and multiply the square by 3 for a defective divisor. 4. 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 ten. 5. Complete the divisor by adding thereto the product of the last figure of the root by the rest and by 30. 6. Multiply and subtract as in simple division, and bring down the next period for a new dividual; for which find a divisor as before, and so proceed till all the periods are brought down. * * *See note under the rule for extracting the square root: it applies equally to this rule. Note.-Defective divisors, after the first, may be more concisely found thus: To the last complete divisor, add the number which completed it with twice the square of the last figure in the root, and the sum will be the next defective divisor. PROOF. Involve the root to the third power, adding the remainder, if any, to the result. EXAMPLES. 1. What is the cube root of 99252.847? 99252.847(46.3 64 Defective divisor & square of 6=4836)35252 +720 complete divisor 5556)33336 Defective divi. & square of 3-634809)1916847 +4140 complete divisor 638949)1916847 Ans. 253. Ans. 73. 2. What is the cube root of 16194277? 4. What is the cube root of 5735339? Ans. 179. Ans. 325. Ans. 280.5 7. What is the cube root of 12.977875? Ans. 2.35 8. What is the cube root of 36155.027576? Ans. 33.06+ 9. What is the cube root of 15926.972504? Ans. 25.16+ 10. What is the cube root of .001906624? Ans. .124. 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. 3000 Ans. 2. A mixed number may be reduced to an improper fraction, or a decimal, and the root thereof extracted. 1. What is the cube root of $48? 2. What is the cube root of 3. What is the cube root of i j o ? 4. What is the cube root of 121? 5. What is the cube root of 31? SURDS. 6. What is the cube root of 7}? 7. What is the cube root of 91? APPLICATION. ? Ans. Ans. 4. Ans. 23. Ans. 1.93+ Ans. 2.092+ 1. The cube of a certain number is 103823; what is that number? Ans. 47. 2. The cube of a certain number is 1728; what number is it? Ans. 12. 3. There is a cistern or vat of a cubical form which contains 1331 cubical feet: what are the length, breadth, and depth of it? Ans. each 11 feet. 4. A certain stone of a cubical form contains 474552 solid inches; what is the superficial content of one of its sides? Ans. 6084 inches. A GENERAL RULE FOR EXTRACTING 1. Point the given number into periods agreeably to the required root. 2. Find the first figure of the root by the table of powers, or by trial; subtract its power from the left hand period, and to the remainder bring down the first figure in the next period for a dividend. 3. Involve the root to the next inferior power to that which is given, and multiply it by the number denoting the given power, for a divisor; by which find a second figure of the root. 4. Involve the whole ascertained root to the given. power, and subtract it from the first and second periods. Bring down the first figure of the next period to the remainder, for a new dividend; to which, find a new divisor, as before; and so proceed. Note. The roots of the 4th, 6th, 8th, 9th, and 12th powers, may be obtained more readily thus: For the 4th root take the square root of the square For the 6th, take the square root of the cube root. For the 8th, take the square root of the 4th root. For the 9th, take the cube root of the cube root. For the 12th, take the cube root of the 4th root. Alligation is a rule for adjusting the prices and simples of compound quantities. CASE 1. To find the mean price of any part of the composition, when the several quantities and their prices are given. P |