64 164)800 656 1688)14400 13504 16965)89600 84825 169702)477500 339404 169704)138096 135763
After getting half of the decimal places, work by contracted division for the other half; and obtain them with the same accuracy as if the work had been at large.
3d, Required the square root of -2916.
-2916(-54 root.
25 104)416
If the square root of a vulgar fraction be required, find the root of the given numerator for a new numerator, and find the root of the given denominator for a new denominator. Thus the square root of $\frac{3}{4}$ is $\frac{3}{2}$, and the root of $\frac{12}{13}$ is $\frac{12}{13}$; and thus the root of $\frac{4}{5} = \frac{4}{5}$ is $\frac{4}{5}$.
But if the root of either the numerator or denominator cannot be extracted without a remainder, reduce the vulgar fraction to a decimal, and then extract the root, as in Ex. 3d, above.
SECT. II.—EXTRACTION OF THE CUBE ROOT.
Rule I.—Divide the given number into periods of three figures, beginning at the right hand in integers, and pointing toward the left. But in decimals, begin at the place of thousands, and point toward the right. The number of periods shows the number of figures in the root.
II.—Find by the table of powers, or by trial, the nearest lesser root of the left-hand period; place the figure so found in the quot; subtract its cube from the said period; and to the remainder bring down the next period for a dividual or resolvent.
The divisor consists of three parts, which may be found as follows: III. The first part of the divisor is found thus: Multiply the square of the quot by 3, and to the product annex two cipher; then inquire how often this first part of the divisor is contained in the resolvent, and place the figure denoting the answer in the quot.
IV. Multiply the former quot by 3, and the product by the figure now put in the quot; to this last product annex a cipher, and you have the second part of the divisor. Again, square the figure now put in the quot for the third part of the divisor; place these three parts under one another as in addition, and their sum will be the divisor complete.
V. Multiply the divisor thus completed by the figure last put in the quot, subtract the product from the resolvent, and to the remainder bring down the following period for a new resolvent, and then proceed as before.
Note 1. If the first part of the divisor happen to be equal to or greater than the resolvent, in this case place 0 in the quot, annex two cipher to the said first part of the divisor, to the resolvent bring down another period, and proceed to divide as before.
Note 2. If the product of the quotient figure into the divisor happen to be greater than the resolvent, you must go back, and give a lesser figure to the quot.
Note 3. If, after every period of the given number is brought down, there happen at last to be a remainder, you may continue the operation by annexing periods of three cipher till there be no remainder, or till you have as many decimal places in the root as you judge necessary.
Ex. 1st. Required the cube root of 12812904.
Cube number 12812904 (234 root.
1st part 1200 4812 resolvent.
2d part 180
3d part 9
I divisor 1369 × 3 = 4167 product.
1st part 158700 645904 resolvent.
2d part 2760
3d part 16
2 divisor 161476 × 4 = 645904 product.
Proof.
| Square | Cube | |--------|------| | 234 | 54756 | | 234 | | | 936 | 219024 | | 702 | 164268 | | 468 | 109512 |
Square 54756 Cube 12812904
Ex. 2d. Required the cube root of 283.
28750000 (306 root.
27
1750000 resolvent.
Divid. 275436 × 6 = 1652616 product.
97384 remainder.
Proof.
| Square | Cube | |--------|------| | 306 | 93636 | | 306 | | | 1836 | 561816 | | 918 | 250908 |
Square 93636 Cube 28750000
If the cube root of a vulgar fraction be required, find the cube root of the given numerator for a new numerator, and the cube root of the given denominator for a new denominator. But if the root of either cannot be extracted without a remainder, reduce the vulgar fraction to a decimal, and then extract the root.