1. What is the cube root of the number 12487168? Ans. 232. 12-487-168(232 Ans 8 A CONCISE METHOD OF EXTRACTING THE CUBE ROOT. RULE. 1.-Point every third figure of the cube given, beginning at the unit's place, then find the highest cube to the first point, and subtract it therefrom, put the root in the quotient, bring down the figures in the next point to the remainder for a resolvend. 2.-Square the quotient, and triple the square for a divisor. As 4x4x3=48. Find how often it is contained in the resolvend, rejecting units and tens, and put the answer in the quotient. 3.-Square the last figure in the quotient, and put it on the righthand of the divisor.-As 6×6=36 put to the divisor 48-4836. 4.-Triple the last figure in the quotient, and multiply by the former, put it under the other, units under the tens, add them together, and multiply the sum by the last figure in the quotient, subtract that product from the resolvend, bring down the next point, and proceed as before. 1. What is the cube root of 99252847? Square of 4x3=48 divisor Square of 6 put to 48-4836 6×3×4 72 5556×6= Square of 46=2116x3=6348 divisor. Square of 39 put to 6348=+634809 3×3×46= 414 638949X3=1916847 2. Required the cube root of 389017. 64 99252847(463 35252 33336 1916847 9. Required the cube root of Ans. 73. Ans. 2805. 10. + When the quotient is 2 or 3, there must be a cypher put to supply the place of tens. When the given number consists of a whole number and decimal together, make the number of decimals consist of 3, 6, 9, &c., places, by adding cyphers thereto, so that there may be a point fall in the unit's place of the whole number. To extract the Cube Root of a Vulgar Fraction. RULE. Reduce the Fraction to its lowest terms, then extract the cube root of its numerator and denominator for a new numerator and denominator; but if the fraction be a surd, reduce it to a decimal, and then extract the root from it. 22. What is the cube root of 4? 23. What is the cube root of ? 24. What is the cube root of ? To extract the cube root of a mixed number. Ans. Ans. Ans. Ans. ·829.+ Ans. ·822.+ Ans. 873.+ RULE. Reduce the fractional part to its lowest terms, and then the mixed number to an improper fraction, extract the cube roots of the numerator and denominator for a new numerator and denominator; but if the mixed number given be a surd, reduce the fractional part to a decimal, annex it to the whole number, and extract the root therefrom. 25. What is the cube root of 1219? 26. What is the cube root of 31? 27. What is the cube root of 40528? Ans. 21. Ans. 34. Ans. 73. 1. If a cubical piece of timber be 47 inches long, 47 inehes broad, and 47 inches deep-how many cubical inches doth it contain? Ans. 103823. 2. There is a cellar dug that is 12 feet every way in length, breadth, and depth-how many solid feet of earth were taken out of it? Ans. 1728. 3. There is a stone of a cubic form, which contains 389017 solid feet-what is the superficial content of one of its sides? Ans. 5329. Between two numbers given, to find two mean proportionals. RULE. Divide the greater extreme by the less, and the cube root of the quotient, multiplied by the less extreme, gives the less mean; multiply the said cube root by the less mean, and the product will be the greater mean proportional. 4. What are the two mean proportionals between 6 and 162 ? RULE. The cube root of the solid content of any solid body given, is the side of a cube of equal solidity. 6. If the solid content of a globe be 10648-what is the side of a cube of equal solidity. Ans. 22. The side of a cube being given, to find the side of the cube that shall be double, treble, &c. in quantity to the cube given. RULE. Cube the side given, and multiply it by 2, 3, &c., the cube root of the product is the side sought. 7. There is a cubical vessel, whose side is 12 inches, and it is required to find the side of another vessel that is to contain three times as much? Ans. 17.307 EXTRACTION OF THE BIQUADRATE ROOT. To extract the biquadrate root is to find out a number, which being involved four times into itself, will produce the given number. RULE. First extract the square root of the given number; and then extract the square root of that square root, and it will give the biquadrate root required. Ans. 531441. Ans. 33362176. Ans. 5719140625. 1. What is the biquadrate of 27 ? 2. What is the biquadrate of 76 ? 3. What is the biquadrate of 275? 4. What is the biquadrate root of 531441? Ans. 27. 5. What is the biquadrate root of 33362176? Ans. 76. 6. What is the biquadrate root of 5719140625. Ans. 275. A RULE FOR EXTRACTING THE ROOTS OF ALL POWERS. 1. PREPARE the number given for extraction, by pointing off from the unit's place, as the root required directs.—2. Find the first figure in the root by the table of powers, which subtract from the given number.-3. Bring down the first figure in the next point to the remainder, and call it the dividend.-4. Involve the root into the next inferior power to that which is given, multiply it by the given power, and call it the divisor.-5. Find a quotient figure by common division, and annex it to the root; then involve the whole root into the given power, and call that the subtrahend.-6. Subtract that number from as many points of the given power as are brought down, beginning at the lowest place, and to the remainder bring down the first figure of the next point for a new dividend.-7. Find a new divisor, and proceed in all respects as before. 1. What is the square root of 141376? SIMPLE INTEREST. There are five letters to be observed in Simple Interest, viz- P. the principal. T. the time. I. the Interest. A. the amount. The ratio is the Simple Interest of £1 for 1 year, at the rate per cent. proposed, and is found thus:-As 100: 3 :: 1:03. As 100 35 :: 1: 035. When the principal, time, and rate per cent. are given to find the interest. RULE. Multiply the principal, time, and rate together, and it will give the interest required. I. When p, r, t, are given to find i. -RULE. prti. 1. What is the interest of £945 10s. for 3 years at 5 per cent., per annum? 945 5X 05X3=141·825, or £141 16s. 6d. Ans. 2. What is the interest of £547 14s. at 4 per cent. per annum, for C years? 3. What is the interest of £796 for 5 years? 4. What is the interest of £397 per aunum? 5. What is the interest of £554 4 per cent. per annum? 6. What is the interest of £236 5 per cent. per annum ? RULE. Ans. £131 8s. 11d. 2:08 qrs. 15s. at 4 per cent. per annum, Ans. £179 5s. 4d. 2 qrs. 9s. 5d. for 2 years at 3 per cent. Ans. £34 15s. 6d. 3.5499 qrs. 17s. 6d. for 3 years, 8 months, at Ans. £91 11s. 1d. 2 qrs. 18s. 8d. for 3 years, 8 months, at Ans. £47 15s. 7d. 2.293 qrs. When the interest is for any number of d ys only. Multiply the interest of £1 for 1 day at the given rate, by the principal and number of days, it will give the answer. INTEREST OF £1 FOR ONE DAY. NOTE.-The above Table is thus found. -As 305 03 :: 1: 00008219178. And as 365 035 :: 1: 00009589041, &c. 7. What is the interest of £240 for 120 days at 4 per cent per an num ? 00010958904×240×120=13 3s. 14d. Aks. J |