3. Find the square root of 1296, and prove. 4. Find the square root of 2916, and prove. 5. What is the value of 7625? Prove. 6. What is the value of Ņ2025? Prove. c. To find the Square Root, when it consists of an integer of more than two figures, fc. 7. Find the square root of 105625, and prove. 105625(3 Root of tens of 30%= 900 2 Root of units, tens. Divisor 2.30=60)156 32 Root of tens. 2:30•2= 120 5 Root of units. 1. 4 325.325=105625 Pr. Divisor 2.320=640)3225 By 10th prin. 1. 2.320.5= 3200 { 52 25 [The method of finding a square root of a larger number of figures does not differ from that of the preceding case, except in slight changes in the numeration of the ranks of the root. See Numeration, p. 117, 1. 1. For instance, the 3 and 2 found at the beginning of the process, are at first considered as the root of tens and units OF TENS; but, as soon as they are found, they are taken together as the tens of the root ; and bringing down to the remainder [32] the first period [25], making 3225, we proceed to develop the root of the true units [5] as before. When there is a remainder after all the periods have been used, it shows that the number whose root is sought is an imperfect power. But we may find a number as near as desirable to the root by annexing periods of ciphers to the remainder, and thence developing roots for the tenths, hundredths, &c., always remembering, however, that all periods for finding square roots, after the first, consist of two figures; and also that we must always begin to mark off both ways at the place of units. Should a remainder occur at the close of the whole process, it must of course be added in, if the work is proved by involution. The following example exhibits a case of this nature : 8. Find the square root of 79238, and prove. Root. of tens. 2:20.8= 320 1 Units. 1. 64 281 - Units. s 28164 Approximate root. 1 Divisor 2.2810=5620)27700 2•2810•4= 22480 By 10th prin. 1. { 16 Remainder, 52.04 42= Proof 281-42 +52.04=79238. 9. Find the square root of 61504, and prove. 10. Find the square root of 43264, and prove. 11. Find the square root of 61928, to two places of decimals, and prove. 12. Find the square root of 363729-61, and prove. 15. What is the value of Ņ2, carried to three places of decimals? Prove. 16. Find the square root of 10 carried to two places of decimals, and prove. EXTRACTION OF THE CUBE ROOT, OR DIVISION INTO THREE EQUAL FACTORS. a. Pointing off Cubes into periods of three figures. Exemplifications for the Black-board. 1. Write in columns, as follows, on the slate or black-board, the cubes of .01, -1, 1, 10, 100, &c., being the smallest significant figure, and the cubes of '09, 9, 9, 90, 900, &c., being the greatest significant figure. Write, also, a line of ciphers, and point them off into periods of three figures each, as under: Cube of 01 of 1 of 1 of 10 TABLE OF CUBES. Cube of .09 of 9 of 900 of 100 = 000000000000000-000000 Suggestive Questions, to be repeated till all can be answered without hesitation.—Of how many figures does the cube of any number of units consist? See 1 and 9 of table of cubes above. Ans. Not less than nor more than Which period will the cube of units occupy, then? How many ciphers are there in the cube of any number of tens? See the cube of 10 and of 90 above. Which period, then, will be occupied by the significant figures of the cube of tens? Which period will be occupied by those of the hundreds ? Of the thousands ? &c. Which period will be occupied by those of the tenths ? Ans. The period to the right of that of the Which period by those of the hundredths ? Where, then, should you look for roots of the units ? For roots of the thousandths? Of the tenths ? Of the tens? Of the hundreds ? &c. Why do we divide numbers whose roots are sought, into periods of three figures ? 6. To find the Cube Root when it contains an integer of two figures. 2. Find the cube root of 13824, and prove. 13824(2 20 8000 4 [Proof 24.24.24=243=13824 Divisor, 3.20=1200)5824 Dividend. 3.202.4 = 4800 By 10th prin., 2.3.4.20= 960 43 64 Suggestive Questions.—What is the greatest cube in 13? What is its root? Is this root 2 units or 2 tens ? Deducting, then, the cube of 20 [8000] from the given cube, what should be looked for next in order in the remainder [5824] ? See the exemplification, in Involution, No. 1, cube of 24; or see the 10th principle, p. 166. Which of these numbers is now known? Ans. Three times the square of If a product [5824) and one of its factors [3.204] be found, how can the other factor be found ? See p. 57, 5. What should be looked for next in the remainder [5824] ? Is it known? What, lastly, will be found in that remainder ? Is 13,824 an exact cube, then ? 3. Find the cube root of 373248, and prove by involution. 4. Find the cube root of 19683, and prove by involution. 5. Find the cube root of 262144, and prove by involution. 6. Find the cube root of 166375, and prove by involution. C. To find the Cube Root, when it contains an integer of more than two figures, foc. The directions given in treating of the extraction of the square root, when it consists of more than two figures, apply almost literally to that of the cube root, namely: Proceed with the two periods at the left, as if these were the whole, and then, bringing down another period, consider the two figures of the root that are thus found as the tens of the root, and find the units of the root as before. Should a remainder occur at the close, annex three ciphers as a period for tenths, unless there should be decimal places sufficient, and proceed to find the root for the rank of tenths, in the same manner that the root for units was found, and so on, as far as may be considered necessary. The evolution of decimal fractions, in fact, does not differ from that of integers. The following example will make all this sufficiently plain. 7. Find the cube root of 14513286-7, and prove. 14513286-7(2 203— 8000 4 Divisor, 3.20%=1200 16513 24 3.202.4= 4800 3 By 10th principle, 2.3.4.20= 960 243 43 64 9 Divisor, 3.2402=172800 1689286 2439 3.2402-3= 51840 By 10th principle, 2.3.3.240= 6480 33 27 3.24302.9= 159432300 By 10th principle, 2. 3.92.2430 590490 93 729 4356.181 Proof, 243-99+4356-181=14513286-7. 8. Find the cube root of 84604519, and prove. 9. Find the cube root of 21024576, and prove. 10. Find the cube root of 28913245, to two places of decimals, and prove. 11. Find the cube root of 21036589, to two places of decimals, and prove. 12. Find the cube root of .000729, and prove. 13. Find the cube root of 2 to two decimal places, and prove. 14. Find the cube root of .02 to two decimal places, and prove. 15. Find the cube root of 20 to two decimal places, and prove. 16. Find the cube root of 3932586^4 to two decimal places, and prove. |