5556 X 6 = 33336 Square of 46-2116 × 3=6348, divisor Square of 39 put to 6348=*634809 3×3×46= 414 638949 x 3 1916847 1916847 2. What is the cube root of 389017 ? 3. What is the cube root of 5735339 ? 4. What is the cube root of 32461759 ? 5. What is the cube root of 84604519 ? 6. What is the cube root of 259694072 ? 7. What is the cube root of 48228544? 8. What is the cube root of 27054036008 ? 9. What is the cube root of 22669810125 ? 10. What is the cube root of 122515327 232 ? 11. What is the cube root of 219365327791 ? 12. What is the cube root of 67 337 3097 125 ? Ans. 73. Ans. 179. Ans. 319. Ans 439. Ans. 638. Ans. 364. -Ans. 3002. Ans. 2805. Ans. 4968. Ans. 6031. Ans. 8765. When the given number consists of a whole number and decimal together, make the number of decimals te consist of 3, 6, 9, &c. places, by adding cyphers thereto, so that there may be a point fall on the unit's place of the whole number. 13. What is the cube root of 12,077875 ? Ans. 2,35 Ans. ,124 * When the quotient is 1, 2, or 3, there must be a cypher put to supply the place of tens. To extract the cube root of a vulgar fraction. RULE. Reduce the fraction to its lowest terms, then extract the cube root of the 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. To extract the cube root of a mixed number. 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. ....... EXAMPLES. 25. What is the cube root of 12? 26. What is the cube root of 31 15,? 27. What is the cube root of 405? SURDS. 28. What is the cube root of 73 ? 29. What is the cube root of 9!? 30. What is the cube root of 8? contain ? THE APPLICATION. Ans. 23. Ans. 3. Ans. 7. Ans. 1,93 + Ans. 2,092 + Ans. 2,057 + 1. If a cubical piece of timber be 47 inches long, 47 inches broad, and 47 inches deep, how many cubical inches doth it Ans. 103823. 2. There is a cellar dug that is 12 feet every way, in length, breadth, and depth, how many solid fect of earth were taken out of it ?. Ans. 1728. 3. There is a stone of a cubic form, which contains 389017 solid fect, what is the superiicial content of one of its sides? Ans. 5329. Between two numbers given, to find two mean proportionals. RULE. Divide the greater extreme by the lesser, and the cube root of the quotient multiplied by the lesser extreme gives the lesser mean; multiply the said cube root by the lesser mean, and the product will be the greater mean proportional. EXAMPLES. 4. What are the two mean proportionals between 6 and 162? Ans. 18 and 54. 5. What are the two mean proportionals between 4 and 108? Ans. 12 and 36. To find the side of a cube that shall be equal in solidity to any given solid, as a globe, cylinder, prism, cone, &c. RULE. The cube root of the solid content of any solid body given is the side of the cube of equal solidity. EXAMPLE. 6. If the solid content of a globe is 10648, what is the side of a cube of equal solidity? Ans. 22. The side of the cube being given, to find the side of that cube, that shall be double, treble, &c. in quantity to the given cube. RULE. Cube the side given, and multiply it by 2, 3, &c. the cube root of the product is the side sought. EXAMPLE. 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. 17,306. 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, then extract the square root of that square root, and it will give the biquadrate root required. EXAMPLES. 1. What is the biquadrate of 27 ? 2. What is the biquadrate of 76 ? 3. What is the biquadrate of 275 ? Ans. 531441. 33362176. 5719140625. 4. What is the biquadrate root of 531441 ? 27. 76. 275. A GENERAL RULE FÖR 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 is 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. X 2 |