To find the cube root of a vulgar fraction, reduce it to its lowest terms, and extract the root of each term separately for the fractional root; but if this cannot be done exactly, make it into a decimal, and extract its root. 12. What is the of? 13. What is the cube root of 1829? 14. What is the cube root of 57819? 15. What is the cube root of 496283.486? 16. What is the cube root of 4896?? Ans. 79.17+ Ans. 16.98+ Ans. .893+ Ans. 23. How do you find the cube root of a vulgar fraction? 17. What is the cube root of 674833 ? Ans. 40.71+ 18. Find the length, breadth, and thickness of a block of wood, of a cubical form, which contains 32768 solid inches. Ans. 32 inches. 19. In digging a cellar, 4628 cubic feet of earth were thrown out; can you tell its size? Ans. 16.66 ft.+ each way. 20. A cubical box contains 15625 round balls, placed in exact order; how many balls in one row? Ans. 25 balls. 21. In a cubical box there are 9261 solid inches; can you tell how many square inches there are in one of its sides? Ans. 441. A RULE FOR EXTRACTING THE ROOTS OF ALL POWERS. 1.-Divide the given number into periods of as many figures as are designated by the required root, beginning at the unit's place. 2.-Find, by the table, or by trial, the first root figure, and subtract its power from the first period. 3. Bring down one figure from the next period, for a partial dividend. 4.-Divide this dividend by the product of the root, involved to a power one short of the given power, multiplied by the given index, for the next root figure. 5.-Draw a line under the last dividend, and bring down two periods; from which subtract the product of the whole ascertained root, involved to the given power. 6. Repeat the former process for the next root figure; and so on, until you have brought down all the periods. If at last you have a remainder, annex ciphers, etc. NOTE.-If the index of the given power be a composite number, you may extract the root, as denoted by its factors; thus, for the 4th root, extract the square root of the square root; for the 6th root, extract the cube root of the square root. What is the rule for extracting the roots of all powers-1st? 2d? 3d? 4th? 5th? 6th? What must be done if you have a remainder? What is said in the note? EXAMPLES. 1. Find the biquadrate or 4th root of 31713911056. 42 × 42 × 42 × 42=3111696 42×4=296352) 596951 the second dividend. Ans. 422. Proof. 4224-31713911056 ✓178084422. 3. Find the 6th root of 7558269224.026249. 4. Find the 4th root of 11788.83463696. 5. Find the 8th root of 78310985281. Ans. 35 Ans. 44.3. Ans. 10.42. 6. Find the 3d and 9th root of 46411484401953. Ans. 23. Ans. The 3d root is 35937; the 9th is 33. 7. Find the ✔, V, V, V, V, V of 48244848604128, and prove the work. MISCELLANEOUS EXAMPLES. 1. A man set out 1296 trees, in an exact square; can you tell how many trees there were in each row? Ans. 36. 2. A dealer sent me a lot of books; the number of leaves in all the books is 14400; the number of leaves in each book, and the whole number of books were equal: can you tell me how many books there were? Ans. 120. 3. A man sold a quantity of cloth for $121.67; the number of pieces, the number of yards in a piece, and the number of cents per yard, were the same: can you tell the whole number of yards sold? Ans. 529. 4. In digging an excavation in the earth, there were thrown out 400 loads of sand; allowing 12 cubic feet for each load, how large must the excavation have been? Ans. 17+ feet long, and the same in width and depth. 5. A company of gentlemen being convened on a certain occasion, one of them says, Come, let us make a contribution for some charitable object. Good, say the rest. Accordingly, the subscription being taken and cast up, it was found that, after paying the doorkeeper his hire, $3, there would be £50 N. Y. currency left. Now, suppose each gentleman subscribed as many shillings as there were gentlemen present; can you tell how many gentlemen there were, and how much each one subscribed ? Ans. There were 32 gentlemen present, and each subscribed $4. Some one may have seen this problem in "The Youth's Cabinet," some years since. The author of this book begs leave to state, that he wrote the original, and sent it to the editor of that paper for publication. DUODECIMAL MULTIPLICATION. DUODECIMAL MULTIPLICATION, or Cross Multiplication, as it is called by some, is in its operation somewhat similar to Compound Multiplication. The denominations are feet, inches, seconds, thirds, fourths, etc.; 12 of the less making 1 of the next greater denomination all through. RULE.-Multiply the multiplicand throughout separately by each denomination in the multiplier, setting down and carrying according to the following table. The sum of the products will be the required answer. TABLE. Feet multiplied by feet give feet. Seconds multiplied by seconds give fourths, etc. ("") Or, add the indices together for the required denomination. EXAMPLES. 1. Multiply 3 ft. 10 in. by 2 ft. 6 in. 2. Multiply 6 ft. 8 in. by 3 ft. 10 in. Ans. 9 ft. 7' 0". Ans. 25 ft. 6' 8". 6. Multiply 12 ft. 6 in. by 10 ft. 4 in. by 8 ft. 6 in. Ans. 1097 ft. 11' 0". 7. Multiply 22 ft. 4 in. by 6 ft. 6 in. by 7 ft. 2 in. Ans. 1040 ft. 4' 4". 8. Multiply 30 ft. 6 in. by 3 ft. 4 in. by 2 ft. 8 in. Ans. 271 ft. 1′ 4′′. 9. Multiply 24 ft. 6 in. by 10 ft. 8 in. by 6 ft. 6 in. 10. Multiply 21 ft. 6' 6" by 3 ft. 3' 3". Ans. 1698 ft. 8'. Ans. 8 ft. 3' 9" 1" 6". 11. Multiply 4 ft. 6' 8" 4" by 2 ft. 4' 3" 2"". Ans. 10 ft. 8' 9" 10" 6" 4' 8". NOTE. It should be borne in mind that Duodecimals, like all compound quantities, can be brought into simple decimals, and the work may be done by that means. MENSURATION. MENSURATION is the art of measuring. In Mensuration, there are three particulars of primary importance; these are length, breadth, and depth, or thickness. Length is the distance from one point or end to another. Breadth is the distance from side to side. Depth, or thickness, is the distance from the top to the bottom. DEFINITIONS. A point has position, but no magnitude. A line has length, but no breadth. Parallel lines are lines whose direction is the same. |