will produce that power. Thus 2 is the square root of 4, because 2x2=4; and 4 is the cube root of 64, because 4x4x4-64, and so on. 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. Separate the given number into periods of two figures, each, beginning at the units place. 2. Find the greatest square contained in the left hand period, and set its root on the right of the given number: subtract said square from the left hand period, and to the remainder bring down the next period for a dividual. 3. Double the root for a divisor, and try how often this divisor (with the figure used in the trial thereto annexed) is contained in the dividual: set the number of times in the root; then, multiply and subtract as in division, and bring down the next period to the remainder for a new dividual. 4. Double the ascertained root for a new divisor, and proceed as before, till all the periods are brought down. Note.-If, when all the periods are brought down, there be a remainder, annex ciphers to the given number, for decimals, and proceed till the root is obtained with a sufficient degree of exactness. Observe that the decimal periods are to be pointed off from the decimal point toward the right hand; and that there must be as many whole number figures in the root, as there are periods of whole numbers, and as many decimal figures as there are periods of decimals. PROOF. Square the root, adding in the remainder, (if any,) and the result will equal the given number. Ans, 327. Ans. 672. 2. What is the square root of 106929? 3. What is the square root of 451584? 4. What is the square root of 36372961? Ans. 6031. 5. What is the square root of 7596796? Ans. 2756.228+ 6. What is the square root of 3271.4007? Ans. 57.19+ 7. What is the square root of 4.372594? Ans. 2.091+ 8. What is the square root of 10.4976? Ans. 3.24 9. What is the square root of .00032754? 10. What is the square root of 10? Ans. .01809 +- To extract the Square Root of a Vulgar Fraction. RULE. Reduce the fraction to its lowest terms, then extract the square root of the numerator for a new numerator, and the square root of the denominator for a new denominator. Note. If the fraction be a surd, that is, one whose root can never be exactly found, reduce it to a decimal, and extract the root therefrom. EXAMPLES. 1. What is the square root of 39qg? 2. What is the square root of 2104? 3. What is the square root of $18? Ans. 3. Ans. .93309+ 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 3736? 2. What is the square root of 279? 3. What is the square root of 8511? 4. What is the square root of 84? APPLICATION. 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 di vidual, 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. |