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to its lowest terms, and extracting the root of the numerator for a new numerator, and of the denominator for a new denominator.

PROOF.

Square the root found, adding in the remainder, if any.

EXAMPLES.

1. Required the square root of 5499025.

5499025(2345 the root.

4

43/149
3129

Proof.

2345 x 2345=5499025

464|2090

4 1856

4685/23425

5 23425 2. Required the square root of 106929. Ans. 327. 3. Required the square root of 152399025. Ans. 12345. 4. Required the square root of 119550669121.

Ans. 345761. 5. Required the square root of 368863.

Ans. 607.34092 + 6. Required the square root of 3.1721812.

Ans. 1.78106+ 7. Huw much is the square root of .00032754?

Ans. .01809+ 8. Required the square root of 794.

Ans.

EXTRACTION OF THE CUBE ROOT. A cube is any number multiplied by its square. To extract the cube root is to find a number, which, being multiplied into itself and then into that product, will produce the given number.

RULE.

1. Distinguish the given number into periods of three figures each, beginning at the unit's place, completing the periods of decimals with ciphers, when necessary.

2. Find the greatest cube in the left hand period, and subtract it therefrom, and put its root in the quotient, and to the remainder annex the next period for a dividend.

CUBE ROOT.

128

3. Square the quotient, and multiply it by 300 for a trial divisor; find how often it is contained in the dividend, and put the result in the quotient.

4. Multiply the figure last put in the quotient, by the rest, and the product by 30; and to the product add the square of the last figure put in the quotient, for the second part of the divisor, and the sum of these complete the divisor.

5. Multiply the complete divisor by the figure last put in the quotient, subtract the product from the dividend, bring down the next period, and proceed as before.

NOTE.—The precepts under the square root are equally applicable here.

PROOF.

Involve the root found to the third power, adding in the remainder, if any; the result will be equal to the given number.

EXAMPLES.
1. Required the be root of 955671625.

9x9x 300=24300*
9x 8 x 30 = 2160

955671625(985 Ans.
8x8
64

729

(

1st Divisor 26524

226671
212192

98 x 98 x 300=2881200*
98 x 5 x 30 14700
5 x 5

25

14479625
14479625

2d Divisor 2895925 2. Required the cube root of 34328125. 3. Required the cube root of 34965783. 4. Required the cube root of 84.604519 5. Required the cube root of 236029032. 6. How much is the cube root of 129?

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A GENERAL RULE FOR EXTRACTING THE

ROOTS OF ALL POWERS. 1. Prepare the given number for extraction, by pointing off from the units, as the required root denotes; thus, for the

square root two figures, cube root three, the fourth root four, &c.

* 24300 the first trial divisor, and 2881200 the second do. do.

2. Find the first figure of the root by trial or inspection into a table of powers, and subtract its power from the left hand period of the given number.

3. To the remainder bring down the first figure in the next period, and call it a dividend.

4. Involve the root, already obtained, to the next inferior power (one less power) to that which is given, and multiply it by the number denoting the given power, for a divisor.

5. Find how many times the divisor may be had in the dividend, and the quotient will be another figure of the root.

6. Involve the whole root to the given power, and sub. tract it always from as many periods of the given number as you have found periods in the root.

7. Bring down the first figure of the next period to the remainder for a new dividend, to which find a new divisor, as before, and in like manner proceed till the whole be finished.

8. If remainders should occur, the root may be continued at pleasure, by annexing ciphers to the given powers.

NOTE 1.-There must be as many whole numbers and decimals, respectively, in the root as there are periods of whole numbers and decimals in the power.

NOTE 2.--Any two sides of a right-angled triangle being given, the third side may be readily obtained, as the square of the hypothenuse, or longest side, equals the sum of the squares of the other two sides, and consequently the difference of the squares of the longest side and either of the other sides, is the square of the remaining side.

Note 3.-When the base and perpendicular, or the two shortest sides, are given, to find the hypothenuse, or longest side. RULESquare the two sides, from their sum extract the square root, and it will be the hypothenuse, longest side.

The longest side and one of the shortest being given, to find the other. Rule-From the square of the longest side, subtract the square of the other given side, and the square root of the remainder will be the side required.

EXAMPLES.

1. What is the square root of 4096 ?

Operation. 4096(64 the root, or ans. 6x6=36

square or 2d power of the quotient 6x2=12)49 Ist dividend 64 X 64=4096 square of the quotient

130 EXTRACTION OF THE ROOTS OF ALL POWERS.

2. What is the cube root of 135796744 ?

135796744(514 root, ans.
5x 5 x 5=125 cube of the quotient

5x5x3=75)107 =

1st dividend

51 x 51 x 51=132651

cube of the quotient

51 x 51 x 3=7803)31457= 2d dividend

514 x 514 x 514=135796744 cube of the quotient

3. What is the biquadrate or 4th root of 19987173376?

19987173376(376 root 3x3x3x3=81=4th power of the quotient

3x3x3x4=108)1188=1st dividend

37% 3737% 37=1874161=4th power of quo.

37 x 37 x 37 x 4=202612)1245563=2d dividend

Ans.

376 x 376 x 376 x 376 =19987173376=4th power of

[the root 4. What is the 6th root of 782757789696 ? Ans. 96. 5. Required the square root of 2916

Ans. 54. 6. What is the cube root of 15625?

Ans. 25. 7. Required the square root of 133225. Ans. 365. 8. I require the square root of 5. Ans. 2.23606+ 9. What is the square root of ?? 10. What is the square root of .00032754? Ans. 018097 11. What is the cube root of 1092727? Ans. 103. 12. What is the cube root of 13. I require the 7th root of 21035.8 Ans. 4.145392

14. A general has an army of 4096 men-how many must he place in rank and file to form them into a square ?

Ans. 64. 15. If 1936 trees be planted in a square orchard, how y must be in a row each way?

Ans. 44.

Ans. ú.

16. Two vessels sail from the same port; one goes due north-30 leagues, and the other due west 40 leagues-how far are they from each other?

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a

17. A line of 50 feet in length extends from the top of wall to a point 40 feet from its base—what was its height?

Ans. ry feet.

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