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the Excefs is not an Unit in the laft Place; for if there were made a fecond Operation, the Root would be 4071,78 c. as may be eafily tried,

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EXAMPLE 3.

Let it be required to extract the Cube Root out of this Number;
Viz. 976379602989073960279630298890

The nearest Cube to 976 is 1000 whofe Root is 10 more than juft,
it's Cube 1000000000000000000000000000000

Remains

976379602989073960279630298890 the Refolvend.
23620397010926039720369701110

The firft Root 10000000000 x 330000000000 the Divifor
Then 30000000000) 23620397010926039720369701110 (78734
6567030867990 for a new Refolvend.

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867990 (

10000000000 = 1ft Root, 787346567030867990 0079364 &c. fubtract. Remains 9920636000 the Root true to the fixth Figure, and only too little by an Unit at the feventh, at the first Operation.

3 Div. 98417) -358756

79

695 I

92246

93

88659

36

295251

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For a fecond Operation (if you require no more than ten Places of Figures true in the Root) you need only affume 9920000000; which being less than juft, proceed with as follows:

From the given Refolvend = 976379602989073960279630298890 Sub. the Cube of 9920000000=976191488000000000000000000000 18811498907396 &c.

Remainder,

Then 3x992 &c. 2976 &c.)18811498907396 &c. (6321068181&c. for a new Refolvend.

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In the fame manner the Cube Roots of Decimal Parts; or of Vulgar Fractions, being firft changed into Decimals, may be extracted.

Sect. 4.

To Extract the Biquadrat Root.

IN extracting the Biquadrat Root, or that of the Fourth Power; (and indeed the Roots of all even Powers) there are some small Difficulties, not fo eafily expreffed and explained in a few Words, as they are by an Algebraick Theorem (fuch as fhall be fhewed further on) I have therefore in this Place made Choice of extracting fuch Roots by two feveral Extractions; and the rather, because I prefume the Reader by this Time thoroughly acquainted with the Bufinefs of extracting the Square Root, by which this may eafily be performed. Thus:

First, Extract the Square Root of the propofed Refolvend, then the Square Root of that firft Root will be the Biquadrat Root required.

Example 1. What is the Biquadrat Root of 4857532416? Firft extract it's Square Root,

Thus

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Then 69696 { being the firft Root, whofe Square Root

4

must now be extracted.

29696 Remainder to be divided by 2.

14848 (264 the Biquadrat Root as was required. 138.

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4.

1048

264

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1048

This is fo eafy I need not infert any more Examples."

Sect. 5. To Extract the Surfolid Root.

HAVING pointed the given Refolvend according as it's Index

denotes; viz. into Periods of five Figures; feeking fuch a Surfolid Number in the Table of Powers (or otherwife) as comes the nearest to the firft Period of the Refolvend, whether greater or lefs; and call it's refpective Root accordingly; viz. more than Juft; or less than Juft; annexing fo many Cyphers to it, as there are remaining Periods of whole Numbers in the Refolvend; as before in extracting the Cube Root: Then find the Difference between the Refolvend, and the Surfolid Number fo taken, by fubtracting the leffer from the greater (as before in the Cube). Next find the Cube of the aforefaid Surfolid Root with it's annexed Cyphers (which you may alfo do by the Table of Powers) and multiply that Cube with 5 the Index of the Surfolid, the Product muft, be a Divifor, by which the Difference between the Refolvend and the Surfolid Number must be divided; that fo it may be depreffed

to

to a Square (as before in the Cube) which must be pointed into Periods of two Figures each, calling it the new Refolvend (as before). Then make the firft Root, without it's Cyphers, a Divifor, enquiring how oft it may be found in the firft Period of the new Refolvend, with this Confideration, if the Root (now a Divifor) be less than Juft, you muft annex twice the Quotient Figure to it; but if it be more than Juft, you must subtract twice the Quotient Figure from a Cypher either annexed, or fuppofed to be annexed to that Divifor or Root, multiplying it fo increased or diminished, with the faid Quotient Figure, fetting down their Product, &c. as before. An Example in each Cafe will render it plain and easy.

Example 1. Suppose it be required to extract the Surfolid Root out of this Number 12309502009375.

12309502009375 The Refolvend pointed.

The nearest Surfolid Number to 1230, the firft Period of the Refolvend, is 1024, whofe Root is 4 being less than Juft. Therefore 12309502009375

1024

2069502009375 their Difference.

Next the Cube of 400 is 64000000 per Table, &c. And 64000000 × 5320000000 the Divifor.

Then 320000000) 2069502009375 (6497 &c.

400

First Root +2×10=+20

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17 the Remainder to be rejected.

That is 415 is the Surfolid Root of the given Refolvend, As may be easily tried by involving it to the fifth Power. Viz 415 × 415 × 415 × 415 × 415=1230950209375 the given Refolvend.

Note, Here again the double Quotient Figure ought to be twice added or fubtracted, in the fame Manner as the fingle one was directed for the Cube Root, page 131, and the Operation for the Surfolid Root in thefe two Examples is performed accordingly: contrary "to what was heretofore done by the Author

Example

Example 2. What is the Surfolid Root of 2327834559873 The neareft Surfolid Number to 232 is 243 whofe Root is 3. being more than just.

Therefore 2430000000000

2327834559873

Remains 102165440127 For a Dividend. The Cube of 300 is 27000000 and 27000000 x 5135000000 Then 135000000) 102165440127 (756,7810 new Refolvend.

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Now the Reason why this Root comes out to fo many Places of Figures at the firft Operation; is because the firft Surfolid Number was fo near the Refolvend, &c. As before.

Sect. 6. To

To Extract the Root of the
Square cubed.

THIS may be eafily performed by two Extractions, according

as it's Name denotes. Thus, firft extract the Square Root of the given Refolvend; then extract the Cube Root of that Square Root, and it will be the Root required: That is, it will be the Root of the fixth Power. Or thus, firft extract the Cube Root of the Refolvend; then extract the Cube Root of that Cube Root, and it will be the Root required.

EXAMPLE 1.

Let it be required to extract the Square cubed Root out of this Number 145220537353515625 the Refolvend.

First I extract the Square Root of this Refolvend, which I take to be the best and cafieft Way. T

Thus

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