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17328=3aa"

228=3a

173508(5218.947 Resolvend.

Divis. 173508=3aat 3a

(7) The next figure in the Root, viz. 3, found as before is also called e; then again 3ade +3eea +eee is the other Subtrahend, or number to be subdueted ; thus, 17328=3aa

3= 3=e

eee viz. 327 3

[blocks in formation]

EXAMPLES. 1 What is the Cube of 6.4 ? Answer 262. 144 2 What is the Cube of .13? Answer .002197 3 What is the Cube of 41. 1 ? Answer 69426.531 4 What is the Cube of .09 ? Answer .000729 5 What is the Cube of .007 ? Answer .000000343 6 What is the Cube Root

Answ. 19.67 + of 7612.812161 ? 7 What is the Cube Root

Answ. 196.71+ of 7612181.77612? 8 What is the Cube Root

Answ. 39.41+ of 61218.00121? 9 What is the Cube Root

Answ. 19.238 + of 7121.1021698 I.

19. What is the Cube Root

11 What is the Cube Root

Answ. .4957 of .121861281 12 What is the Cube Root

Answ. .19 107 + of .0069761218?

13 If a cubical piece of timber be 41 inches long, 41 inches broad, and 41 inches deep, how many cubical inches doth it contain ? Ans. 68921 cubical inches.

14 Suppose a Cellar to be dug, that shall be 12 feet every way.in length, breadth and depth ; how many solid feet of earth must be taken out to eoinplete the same? Ans. 1728.

15 Suppose a stone of a cubic form to contain 474552 solid inches; what is the superficial content of one of its sides ? Ans. 6084 inches.

OF THE CUBE-ROOT OF A VULGAR FRACTION. Q. How do you extract the Cube Root of a Vulgar Fraction ?

A. 1 Reduce the Fraction to its lowest terms.

2 Extract the Cube Roots of the Numerator and Denominator for a new Numerator and Denominator.

3 If the Fraction be a Surd reduce it to a Decimal, and then extract the Cube Root from it.

4 The Decimal Fraction must consist of Ternaries of places; as three, six, nine, &c.

EXAMPLES. 1 What is the Cube Root of $? Answer je 2 What is the Cube Root of 123? Ans. I 3 What is the Cube Root of 1466 ? Ans.

SURDS. 4. What is the Cube Root of ? Ans. .763+ 5 What is the Cube Root of q? Ans. 9494 6 What is the Cube Root of ? Ans. 693+

OF THE CUBE ROOT OF A MIXED NUMBER. Q. How do you extract the Cube Root of a mixt number? A. 1 Reduce the fractional part to its lowest terms. 2 Reduce the mixt number to an improper fraction,

3 Extract the Cube Roots of the Numerator and Denominator, for a new Numerator and Denominator.

4 If the mixt number. given be a Surd, reduce the fractional part to a Decimal, and annex it to the whole number, and extract the Cube Root from the whole.

EXAMPLES. 1 What is the Cube Root of 57812?Ans. 8. 2 What is the Cube Root of 421 ? Ans. 3} 3 What is the Cuhe Root of 519 ? Ans. ) 1월

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SURDS. 4 What is the Cube Root of 81, ? Ans 2.013.+ 5 What is the Cube Root of 7? Ans. 1.966+

OF THE BIQUADRATE ROOT. WHAT is the Biquadrate Number? A. Any Number involved four times produces a Biquadrate.

Q. How is the Biquadrate Root extracted ?

A. First extract the Square Root of the giren Resolvond, and then extract the Square Root of that Square Root, for the Biquadrate Root required.

EXAMPLES.

1 What is the Biquadrate of 48 ? Ars. 5308416. 2 What is the Biquadrate of 96 ? Ans.84934656. 3 What is the Biquadrate Root of 5308416 ? Ans. 48. 4 What is the Biquadrate Ropt of 84934656 ? Ans. 96. 5 What is the Biquadrate Root

Ans. 384. of 2174327 1936 ?

OF THE SURSOLID ROOT.

Q. WHAT is a Sursolid ?

"

A. Any Number involved 5 Times produces a Sirsolid.

Q. How is the Sursolid Root, or the Root of any other higher Power extracted ? :

A. By the following general Rules.

If any even power be given, let the Square Root of it be extracted, which reduces it to half of the given Power, then the Square Root of that Power resluces it to half of the same power, and so on till you come to a square of a Cube.

For example: Suppose a 241h Power be given : the Square Root of that reduces it to a 12th Power, the Squaro Root of the 12th Power reduces it to a 6th Power, and the Square Root of the 6th Power to a Cube.

2 If any odd Power he given, as the 17th, fic. observe

[1] From the Unity Place, both ways, point off at every such Number of figures as is the Index of the power for a Resolvend.

[2] Seek in the Table of Powers, for such a power (being the same Power with the Index) as comes nearest the first Period, whether greater or less, calling its Root accordingly more than just, or less than just.

(3) Annex so many Cyphers to the Root, as there are

of the square,

Periods of whole Numbers in the given Resolvend.

[4] Find the difference between the given Resolvend and the power coming nearest the first Period.

[5] Whatever odd power is given, the next lowest odd power to that of the said root must be found with its annexed cyphers ; i. e. if the ninth power be given, find the 7th power of the root and cyphers: if the lith power be given, find the 9th, d'e.

[6] Multiply the next lowest odd power by the Index of the given power, and let that product be a divisor to the difference between the given resolvend and power first found, which depresses it to a square.

[7] Point this square into periods of two figures cach.

[8] Then make the first root without ils cyplier a divisor, and ask how oft it may be found in the first period

[9] If the divisor be less than just, you must multipyi the quotient figure by half the iudex, i e if the index be 11, multiply the quotient figure by 5; if the index be 9, multiply it by 4, d.e. and add it to the divisor; but if it be more than just, you must subtract it from the divisor, having a cypher annexed or supposed to be annexed to the divisor ; which sum or difference must be multiplied by the said quotient figure, and so continued to every pew figure in the quotient.

[10] If the first root with its cyphers be more than just, the quotient must be subtracted from it; but if it be less than just, it must be added to it; and the sum or difference will be the root required.

3. If an even power be giver, and the square root of that power be extracted, reduce it to an odd power ; you must then proceed with that odd power as the foregoing rules direct.

EXAMPLES.
1. What is the sursolid of 6436313 ?

.6436343
32 the nearest sursolid, whose root and cypher is 20.

3236343 The cobe of 20 is =8000 And 8000 X5 is=40000

Then 40000)323634530

Again 2 )80 3 + 3 +2=678

Lastly 20

+3

23 the sursolid

N?

Ist divisor=26

Root required 2 to be rejected. Note. This is a very expeditious way of extracting the roots of high powers, but it is not always exact, because (as Àr. Ward observes, for it was taken from him) there will be à remainder, and sometimes an Excess or Defect in the last Figure of the root, when the given resolvend or power hath a true root-as appears by the fifth example following, whose true root should not be 384.3, as it there stands, but 384.

2. What is the sursolid of 48 ? Ans. 254803968. 3. What is the sursolid root of 8153726976? Ans. 96. 4. What is the sursolid root of 254803968 ? Ans. 48. 5. What is the sursolid root of 2

Ans. 384.3. 8349416423424

OF THE SQUARE-CUBE ROOT. Q. WHAT is a Square Cube ?

A. Any Number involved six times, produces a Square Cube.

EXAMPLES.

1. What is the Square-Cube of 48 ? Aos. 12230590464.

2. What is the Square-cube root of 78275778969.6.? Ans. 96.

3. What is the Square-cube root of 12230590464? Ans. 48

4. What is the Square-cube root of 320617590659 4816? Answer 384.

OF THE SECOND SURSOLID-KUUT. Q. WHAT is the Second Sursolid ?

A. Any number involved seven times produces a second Sursolid.

EXAMPLES. 1. What is the 2d sursolid of 96 ? A. 75144747810816.

2. What is the second sursolid root of 75144747810816? Answer 96.

3. What is the second sursolid root of 587068342272 ? Answer 48.

4. What is the 2d sursolid root of 12311715481324093447 Answer 38 1.42.

OF THE SQUARE BIQUADRATE-ROOT. Q. Wh4T is a Square Biquadrate ? 1. Any number involved eight times, is a Biquadrate squared, or square biquadrate.

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