B

THE EXTRACTION OF THE

CUBEROот.

Y the Extraction of the Cube Root of any Number, is meant the finding out such a Number, which being multiplied by itself, and that Product again multiplied by the Name Number, shall produce the given Number. Thus, Tuppose 27 were given to have its Cube Root extracted, we hall find it to be 3; because 3 multiplied by 3, is 9; and multiplied by 3, is 27; which is equal to the given Num

ber.

And this may be demonstrated thus :

Suppose we take 9 little Cubes, (or Dies) and lay them down, so as to form a Square whose Side shall be 3; upon these let there be laid 9 more Cubes, and upon them let there be laid 9 more; then will there be in all 27 Cubes, which will make one greater Cube, as ABCDEFG,

whose Length, Breadth, and

Depth will be 3 Cubes; and this
greater Cube contains just 27
Tess Cubes; the Extraction there.
fore of the Cube Root is by hav- F
ing the Number of littles Cubes
given which are contained in a
greater Cube; to find how many
of the less Cubes make one Side
of the greater.

E9

D

9

G

27

C

A

9

B

To extract the Cube Root of any Number, it will be necessary to have in Memory these Cubes, whose Roots are one Figure.

Root 123456789 Cube 18 27 64 125 216 343 512 729

In this Table you see the Cube of 1 is 1; the Cube of z is 8; the Cube of 3 is 27; and so of the reft.

Now

Now, the Cube Root of any Number, (greater tha those expressed in the foregoing Table) may be found ou by the following

Rule.

First, having fet down the given Number, or Refolvend make a Dot over the Unit Figure, and so on over ever third * Figure (towards the Left Hand in whole Numbers but towards the Right Hand in Decimals); and fo man Dots as there are, so many Figures will be in the Root.

Next, seek the nearest Cube to the first Period; place its Root in the Quotient, and its Cube set under the firf Period. Subtract it therefrom; and to the Remainder bring down one Figure only of the next Period, which wil be a Dividend.

Then, Square the Figure put in the Quotient, and multiply it by 3, for a Divifor. Seek how oft this Divifor may be had in the Dividend, and set the Figure in the Quotient which will be the fecond Place in the Root.

Now, cube the Figures in the Root, and fubtract it from the two first Periods of the Resolvend; and to the Remain der bring down the first Figure of the next Period, for a new Dividend. Square the Figures in the Quotient, and multiply it by 3, for a new Divifor; then proceed in all Respects as before, till the Whole is finished.

* The Reafon for pointing every third Figure is, because the Cube of the greatest Number under 10, will confift but of three Places.

The following Examples will make this difficult Rule plain and easy.

What is the Cube Root of 13824?

Resolvend. Quotient.

Divifor.

13824 (24 true Root.

8

Square of 2 multiplied by 3 = 12) 58 Dividend.
Subtract Cube of 24 from Resolvend = 13824

(0) Remains nothing.

Having pointed the given Number according to the foregoing Rule, we find there will be two Places in the Root; because there are two Dots in the Resolvend. We then feek the greatest Cube in 13 the first Period, which we find to be 8; whose Root 2, we place in the Quotient for the first Figure of the Root; the 8 we set under the 13, fubtract it therefrom, and to the Remainder 5 bring down the first Figure of the next Period for a Dividend. We next divide the Dividend 58 by three Times the Square of 2 (for a Divifor) which makes 12; and the Quotient 4, arifing from that Divifion, is the second Figure in the Root; the Cube of which whole Root we fubtract from the whole Refolvend, and find the Remainder to be (o) or Nothing. This shows that 24 is the true Root; because its Cube is exactly equal to 13824, the given Number.

To prove in all Cases if the Operation be right; multiply the Root by itself, and that Product again by the Root; to which add the Remainder, if any; then, if that Sum be equal to the given Number, the Work is right; otherwise not; and must be performed over again.

Let it be required to extract the Cube Root out of the Number 13312053?

Square of 2 multiplied by 3 = 12) 53 Dividend

=

Subtract the Cube of 23

12167

Divisor.

Sq. of 23 multiplied by 3=1587) 11450 new Dividend.
Subtract the Cube of 237 =

13312053

(0)

Here the given Number is a true Cube; for when the Cube of 237 is fubtracted from the whole Resolvend, there will remain Nothing.

What is the Cube Root of 48228544?

Square of 3 multiplied by 3=27) 212 Dividend.

Subtract the Cube of 36

= 46656

Divifor.

Sq. of 36 multiplied by 3 = 3888) 15725 new Dividend. Subtract the Cube of 364.

=

48228544
(0)

In this Example also, the Number given is a true Cube; for when the Cube of 364 in the Root, is fubtracted from the given Resolvend, there will be no Remainder.

If the given Number be not a perfect Cube Number, (that is, hath not a Root expreffible exactly by any true Number) we may annex two or three Periods or Cyphers to it; and for every Period so annexed we shall have one Decimal Place in the Root; as in this Example.

What is the Cube Root of 412?

412.000.000 (7.44 Root.

Cube of 7 343

Square of 7 multiplied by 3=147) 690
Subtract the Cube of 74 = 405224

Square of 74 multiplied by 3=16428)67760
Subtract the Cube of 744

=

411830784

(169216) Remainder.

In this Example, two Periods of Cyphers are annexed to the given Number; we have therefore 2 Decimal Places in the Root.

Note. If the Cube of the Root should be greater than the Periods of the Resolvend from which it is to be taken; you must diminish the last Figure put in the Root, till the Cube be equal to, or less, than those Periods of the given Number.