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Ex. 2. Extract the cube root of 571482.19.
II. To extract the Cube Root by a short Way*.
1. By trials, or by the table of roots at p. 90, &c, take the nearest rational cube to the given number, whether it be greater or less; and call it the assumed cube.
* The method usually given for extracting the cube root, is so exeedingly tedious, and difficult to be remembered, that various other approximating rules have been invented, viz. by Newton, Raphson, Halley, De Lagny, Simpson, Emerson, and several other mathematicians; but no one that I have yet seen, is so simple in its form, or seems so well adapted for
general use, as that above given. This rule is the same in effect as Dr. Halley's rational
2. Then say, by the Rule of Three, As the sum of the given number and double the assumed cube, is to the sum of the assumed cube and double the given number, so is the root of the assumed cube, to the root required, nearly. Or, As the first sum is to the difference of the given and assumed cube, so is the assumed root to the difference of the roots nearly.
3. Again, by using, in like manner, the cube of the root last found as a new assumed cube, another root will be obtained still nearer. - And so on as far as we please ; using always the cube of the last found root, for the assumed cube.
To find the cube root of 21035.8.
Here we soon find that the root lies between 20 and 30, and then between 27 and 28. Taking therefore 27, its cube is 19683, which is the assumed cube. Then
60401.8) 1667374.2 (27.6047 the root nearly.
formula, but more commodiously expressed; and the first investigation of it was given in my Tracts, p. 49. The algebraic form of it is this:
As P + 2A : A + 20 :: : R. Or,
As P + 2A : PO.A :: 7; Rr; where p is the given number, A the assumed nearest cabe, r the "cube root of n, and R the root of 2 soughte
Again, for a second operation, the cube of this root is 21035.318645155823, and the process by the latter method will be thus :
diff. •481355 :: 27.6047 :
conseq. the root req. is 27.604910560. Ex. 2. To extract the cube root of .67. Ex. 3. To extract the cube root of .01.
TO EXTRACT ANY ROOT WHATEVER *.
1 times P, 1 times A ;
Let o be the given power or number, n the index of the power, a the assumed power, r its root, R the required root of P. Then
say, As the sum of n + 1 times A and n is to the sum of n + 1 times P and n so is the assumed root r, to the required root R.
Or, as half the said sum of n + 1 times á, and n-1 times P, is to the difference between the given and assumed powers, so is the assumed root r, to the difference between the true and assumed roots; which difference, added or subtracted, as the case requires, gives the true root nearly. That is, n+1. A +-1. p:n+ 1'. P. tn-l. A ::8: R.
Or, ntl. A +1-1. P: Pes A :: 0 : Rr. And the operation may be repeated as often as we please, by using always the last found root for the assumed root, and its nth power for the assumed power A.
* This is a very general approximating rule, of which that for the cube root is a particular case, and is the best adapted for practice, and for memory, of any that I have yet seen. It was first discovered in this form by myself, and the investigation and use of it were given at large in my Tracts, p. 45, &c.
To extract the 5th root of 21035.8.
Here it appears that the 5th root is between 7.3 and 7.4. Taking 763, its 5th power is 20730-71593. Hence we have P= 21035.8, n = 5,=7.3 and A = 20730•71593; then n+1. A +9-1. LP: POR A :: 5 : Ron r, that is, 3 x 20730:71593 +2 21035.8 : 305.084 :: 7.3 : 3
1. What is the 3d root of 2 ? 2. What is the 3d root of 3214? 3. What is the 4th root of 2 ? 4. What is the 4th root of 97.41 ? 5. What is the 5th root of 2 ? 6. What is the 6th root of 21035.8? 7. What is the 6th root of 2? 8. What is the 7th root of 21035.8 ? 9. What is the 7th root of 2 ? 10. What is the 8th root of 21035.8 ? 11. What is the 8th root of 2 ? 12. What is the 9th root of 21035.8 ? 13. What is the 9th root of 2 ?
Ans. 1.259921. Ans. 14.75758. Ans. 1:189207. Ans. 3•1415999. Ans. 1•148699. Ans. 5.254037. Ans. 1.122462. Ans. 4:145392. Ans. 1.104089. Ans. 3.470323. Ans. 1.090508. Ans. 3.022239. Ans. 1.080059.
The following is a Table of squares and cubes, as also the square roots and cube roots, of all numbers from 1 to 1000, which will be found very useful on many occasions, in numeral calculations, when roots or powers are concerned.