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mainder bring down the next Period, which call the Refolvend, and draw a Line under it.

Now fquare the Figure in the Root, and multiply it by 300, putting the Product under the Refolvend; then multiply the Figure in the Root by 30, which place under the laft Product; and the Sum of thefe two Products call a Divifor, under which draw a Line.

Find how many times this Divifor can be had in the Refolvend, and this is the fecond Figure in the Root, which place in the Root.

Multiply this laft Figure in the Root by itself, and place the Product under the Divifor; then multiply that Part of the Divifor, which arifes from multiplying the former Figure in the Root by 30, by the laft Figure in the Root, and put the Product under the laft Product; under these place the other Part of the Divifor, and add them together; and multiply their Sum by the laft Figure in the Root, calling the Product the Subtrahend, and fubtract it from the Refolvend.

If the Root is only to confift of two Places, the Operation. is finished, as in the following Examples.

Exam. 1. To extract the Cube of Root 15625.

15625 (25 Root

Proof.

25 Root.

25

7625 Refolvend.

125

1200 the Product of the Square of 2 by 300. 60 the Product of 2 by 30.

50

625 Square.

1260 Divifor.

25

[blocks in formation]

The firft Period in the given Number is 15; and, by th Table of Cubes, Page 89, the Cube less than 15 is 8, which place under the 15, and 2, the Cube Root of 8, in the Root. The Reader, by comparing the Directions with the Operation, will find it very eafy. To try whether 25 is the exact Cube Root of 15625, multiply 25 by 25, and this Product again by 25; this laft Product is 15625, therefore the true Cube Root of 15625 is 25.

Exam. 2. Extract the Cube Root of 438976.

438976 (76

343

95976 Refolvend.

14700 Product of the Square of 7 by 300.

210 Product of 7 by 30.

14910 Divifor.

36 Product of 6 by 6.

1260 Product of 210 by 6.

14700 the other Part of the Divifor.

15996

6 the Figure laft placed in the Root. 95976 Subtrahend,

o remains,

Exam. 3. Extract the Cube Root of 205379.

[blocks in formation]

Exam. 4. Extract the Cube Root of 18399744:

[blocks in formation]

1200 Product of the Square of 2 by 300.
60 Product of 2 by 30.

1260 Divifor.

36 Square of 6.

360 Product of 60 by 6.

1200 the other Part of the Divifor.

1596

6 the Figure last placed in the Root.

9576 Subtrahend.

823744 Refolvend.

202800 Product of the Square of 26 by 300,

780 Product of 26 by 30.

203580 Divifor.

16 Square of 4.

3120 Product of 780 by 4.

202800 the other Part of the Divifor.

205936

4 the Figure laft placed in the Root.

823744 Subtrahend.

o remains.

Exam. 5.

Exam. 5. Extract the Cube Root of 54 to three Places of Decimals.

[blocks in formation]

418519

7 the Figure laft placed in the Root.

2929633 Subtrahend.

417367000 Refolvend.

42638700 Square of 377 by 300.
11310 Product of 377 by 30.

42650010 Divifor.

81 Square

81 Square of g.

101790 Product of 11310 by 9. 42638700 the other Part of the Divifor

42740574

9 the Figure laft placed in the Root.

384665139 Subtrahend.

32701861 Remainder,

By the two laft Examples the Learner may fee, that, when a new Refolvend is made, all the Figures then in the Root must be squared, and their Square multiplied by 300, and the Figures in the Root multiplied by 30 to make a Divifor: That, when it is known how many times the Divifor can be had in the Refolvend, that Figure must be placed in the Root, to the right Hand of thofe already there, as in the Quotient in Divifion: That then the Figure laft placed in the Root must be fquared, and put down under the Divifor; and the former Part of the Root multiplied by 30 is to be multiplied by this laft placed Figure in the Root, and put under the Square of the laft Figure in the Root; under these put the other Part of the Divifor, which being added together, their Sum must be multiplied by this last placed Figure in the Root, for the Subtrahend to be fubtracted from the Refolvend.

If the Subtrahend, when all the Periods are taken down, is the fame with the Refolvend, the Work is finished, and the Root is the true Root fought; as in the fourth Example, where 264 is the true Cube Root of the given Cube: But, if any Thing remains when all the Periods have been taken down, to that Remainder annex three Cyphers, and pursue the fame Method as before; by which the Root may be con tinued to as many Places of Decimals as may be thought proper; always obferving to annex three Cyphers to each Remainder, to make a new Refolvend.

If a Decimal Fraction, or mixed Number, is given for the Cube Root to be extracted, and the decimal Places are neither three, fix, nor nine, Cyphers must be annexed, to make the Decimals have one of thofe Numbers of Places; fo that the Dot may fall on the Units Place of the integral Part of the mixed Number, and that the firft Period, in an entire Decimal, may confift of three Places.

Exam. 6.

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