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EXAMPLES

What is the cube root of 99252847

99252847(463

64=Cube of 4.

Divisor.
Square of 4x3=48)35252 Resolvend.

216=Cube of 6.
432 =4X3 X by square of 6.
288 =Divisor X by 6.

33336 Subtrahend.

Divisor.
Square of 46 X 3=6348) 1916847 Resolvend.

27=Cube of 3.
1242 =46X3 X by Square of 3.
19044

=Divifor x by 3. 1916847 Subtrahend.

Another new and more concise method of extracting the Cube Root.

RULE. 1. Point every third figure of the cube given, beginning at the unit's place, then find the nearest cube to the first point, and subtract it therefrom, put the root in the quotient, bring down the figures in the next point to the remainder for a resolvend.

2. Square the quotient and triple the square for a divisoras 4X4X3=48. Find how often it is contained in the resolvend, reječting units and tens, and put the answer in the quotient.

3. Square the last figure in the quotient, and put it on the right hand of the divisor :

As 6X6=36 put to the divisor 48=4836. 4. Triple the last figure in the quotient, and multiply by the former, put it under the other, units under the tens, add them together, and multiply the sum by the last figure in the quotient, subtract that product from the resolvend, bring down the next point and proceed as before.

47646

2.

Anf. 73. Ans. 179. Anf. 319. Anf. 439.

Examples
What is the cube root of 99252847 ?

Square of 4X3=48 Divisor. 99252847(463
Square of 6, put to 48=4836

64
6X3 X4=72

35252

5556 X 6 = 33336 Square of 46=2116X3=6348 Divisor Square of 3=9 put to 6348=*631809 1916847

3X3 X 46=414

638949X3=1916847 What is the cube root of 389017? 3.

What is the cube root of 5735339 ? 4. What is the cube root of 32461759 ? 5. What is the cube root of 84604519 ? 6. What is the cube root of 259694072 ?

Ans. 638. 7.

What is the cube root of 48228544 ?
8. What is the cube root of 27054036008 ?
9.

What is the cube root of 22069810125? Anf. 2805.
What is the cube root of 122615327232 ? Anf. 4968.
What is the cube root of 219365327791 ?

Ans. 6031. What is the cube root of 673373097125? Ans. 8765 When the given number consists of a whole number and decimal together, make the number of decimals to consist of

3, 6, 9, &c. places, by adding cyphers thereto, fo that there may be a point fall on the unit's place of the whole number. 13.

What is the cube root of 12.977875 ? 14

What is the cube root of 36155.027576? Ans. 33,06+ 15.

What is the cube root of ,001906624? 16. What is the cube root of 33:230979637 ?

Anf. 3,215+ 17. What is the cube root of 15926,972504?

Anf. 25,19+ is. What is the cube root of ,053258279 ?

Anf. ,376 +

Anf. 364
Ans. 3002

10.

11.

12.

Anf. 2,35

Ans. ,124

To extra&t the Cube Root of a Vulgar Fraction. Rule. Reduce the fraction to its lowest terms, then extract the Gube root of the numerator and denominator for a new numerator and denominator, but if the fraction be a surd, reduce it to a decimal, and then extract the root from it.

EXAMPLES. 19. What is the cube root of ?

Anf. 4.
What is the cube root of 324. ?

Ani,

20.

* When the quotient is 2 or 3 there must be a cypher put to supply the place of tens,

21. What is the cube root of 8?

Ans. ' .

SURDS. What is the cube root of 4 ? 23. What is the cube root of ? 24. What is the cube root of ?

Ans. ,829+
Ans. ,822 to
Anf. ,873+

To extraet the Cube Root of a mixt number. RULE. Reduce the fractional part to its lowest terms, and then the mixt number to an improper fraction, extract the cube roots of the numerator and denominator for a new numerator and denominator ; but if the mixt number given be a surd, reduce the fractional part to a decimal, annex it to the whole number, and extract the root there. from.

EXAMPLES. 25. What is the cube root of 12 ??

Anf. 2. 26. What is the cube root of 31 32 ?

Anf. 3. 27. What is the cube root of 405 15 ?

Anf. 75

SURDS. 28. What is the cube root of 7} ?

Anf. 1.93+ 29. What is the cube root of g ?

Ans. 2,092+ 30. What is the cube root of 8??

Anl. 2,057+ THE APPLICATION. If a cubical piece of timber be 47 inches long, 47 inches broad, and 47 inches deep, how many cubical inches doth it contain ?

Anf. 103823 There is a cellar dug that is 12 feet every way, in length, breadth and depth, how many solid feet of earth were taken out of it ?

Ans. 1728. 3. There is a stone of a cubic form, which contains 389017 solid feet, what is the superficial content of one of its fides ?

1.

2.

Anf. 5329.

Between two numbers given, to find two mean proportionals. Rule. Divide the greater extreme by the lesser, and the cube root of the quotient multiplied by the lesser extreme, gives the lesser meant; multiply the said cube root by the lesser mean, and the produ& will be the greater mean proportional.

Dd

EXAMPLES. 4. What are the two mean Proportionals between 6 and 162?

Anf. 18 and 54. 5. What are the two mean Proportionals between 4 and 108 ?

Anf. 12 and 36.

To find the side of a cube that shall be equal in folidity to any given

folid, as a globe, cylinder, prism, cone, &c. Rule. The cube root of the solid content of any solid body given is the side of the cube of equal folidity.

EXAMPLE. 6. If the folid content of a globe is 10648, what is the side of a cube of equal solidity ?

Ans. 22.

The side of the cube being given, to find the fide of that cube, that shall

be double, treble, &c. in quantity to the given cube. Rule. Cube the fide given, and multiply it by 2,3, &c. the cube root of the product is the side fought.

EXAMPLE 7. There is a cubical vessel, whose fide is 12 Inches, and it is required to find the side of another vessel, that is to contain 3 times as puch ?

Anf. 17,306.

EXTRACTION OF THE .BIQUADRATE ROOT.

To extra&t the Biquadrate Root is to find out a number, which being involved four times into itself, will produce the given number.

Rule, First extract the Square Root of the given number, and then extract the Square Root of that Square Root, and it will give the Biquadrate Root required.

2.

EXAMPLES.
What is the biquadrate of 27 ?

Anl. 531441
What is the biquadrate of 76 ?

33362176 3. What is the biquadrate of 275 ?

5719140625 4. What is the biquadrate root of 531441

27 5. What is the biquadrate root of 33362176 ?

76 What is the biquadrate root of 5719940625 ?

275

14

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1.

2.

PREPARE the number given for extraction, by pointing off from the units' place, as the root required directs.

Find the first figure in the root, by the table of powers, which subtract from the given number.

3. Bring down the first figure in the next point to the remainder, and call it the dividend.

4. Involve the root into the next inferior power to that which is given ; multiply it by the given power, and call it the divisor.

5. Find a quotient figure by common division, and annex it to the root ; then involve the whole root into the given power, and call that the subtrahend.

6. Subtract that number from as many points of the given pow. er as is brought down, beginning at the lowest place, and to the remainder bring down the firit figure of the next point for a new div. idend. 7. Find a new divisor, and proceed in all respects as before.

EXAMPLES.
1. What is the square root of 141376 ?

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