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Prob. VI. 'The sum of any two nunsbess, and their protlucts being given, to find cacli number,

RULE. Froin thic

square of their sum, subtract 4 times their product, and extract the square metode ininter which will be the difference of the two numbers; den

S half the said difference and to the sidig enes the greater of the two numbers, and tuc said bail difference subtracted from the half sam, gives the lesser number.


The sum of two numbers is 43, and their product is 442; what are those two numbers ?

The suin of the numb, 43.X 45=1849 square of do. 'The product of do. 442x 4=1768 4 times the pro. Then to the sum of 21,5

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✓ 81=9. diff. of the

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EXTACTION 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 its square, shall produce the given nuinber.

RULE. 1. Separate the given number into periods of three fig. ures cach, by putting a point over the unit firure and every third figure from the place" of units to the itīt, und if there be decimals, to the right.

2. Find the greatest cube in the left hand periol, and place its root in the quotient.

3. Subtract the cube this found, from the said pcriod, and to the remainder bring down the next period, calling this thie Jiviilent.

4. Multiply the sugare of the quotient by 300, calling it the divisor.

3. Seek how often the divisor may be had in the divi. dad, and place the result in the quotient; then multiply the divisor by this last quotient figurc, placing the product under the dividend.

ô. Multiply the former quotient figure, or figures by the square of the last quotient figure, and that product by 30, and place the product ander the last; then under these two products place the cube of the last quotient figure, and add them together, calling their süm the subtrahend.

7. Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend ; with which proceed in the same manner, till the whole be finishe:l.

NOTE:---If the subtrahend (found by the foregoing rule) happens to be greater than the dividend, and consequent ly camot be subtracted therefrom, you must make the last quotient figure one less; with whicle find a new sub. trahend, (by the rule foregoing) and so on until you can subtract the subtraliend from the dividend.


1. Required the cube root of 18399,744.

19999,744(26,4 Root. Ans. 8

2X254x300=1200)10399 first dividend.

7200 6x6=36X2=72X30=2160

6x6x6= 216

9576 1st subtrahcnd. 26X26=676 X 300=202800) 823744 ad dividend.

811200 4X4=16x26=416x30= 12480

4X4X45 64

823744 2d subtrahend.

Note.-The foregoing example gives a perfect root; and if, when all the periods are exhausted, there happens to be a remainder, you may annex periods of cyphers, and continue the operation as far as you think it necessary.

Answers 2. What is the eube root of 205379 ?

59 3. Of

614125 :

85 f. Of


346 5. Ot

146568,183 ?

52,7 6. Of

29,303629 ? 3,097 7. Of

80,763 ? 4,32+ 8. Of

,162771536 ?

,546 9. Of

,000684134 ?

,088+ 10 Of 122615327 232 ?

4968 RULE II. 1. Find by trial, a cube near to the given number, and call it the supposed cube.

2. Then as twice the supposed cube, added to the given number, is to twice the given number added to the supposed cube, so is the root of the supposed cube, to the true rost, or an approximation to it.

3. By taking the cube of the root thus found, for the supposed cube, and repeating the operation, the root will be liad to a greater degree of exactness.

Let it be required to extract the cube rootwof 2.

Assume 1,3 as the root of the nearest cube, then 1,3 x 133 x1,5=2,197=supposed cube. Then, 2,197 2,000 given numbers 2


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As 6,394 6,197 1,3 1,2599 root, which is true to the last place of decimals; but might by repeating the operation, be brought to greater exactness 2. What is the cube root of 584,277056

Anna 8,36

3. Required the cube root of 729001101 ?

Ans. 900,0004



Showing the use of the Cube Root. 1. The statute bushel contains 2150,425 cubic or solid inches. Idemand the side of a cubic box, which shall contain that quantity ?

52150,425=12,907 inch. Arzs. Note --The solid contents of similar figures are in proportion to cach other, as the cubes of their similar sides or diameters,

2. If a bullet 5 inches diameter, weigh 41). wint will a bullet of the same metal weighi, whose diameter is o inches :

3X3X3=27 6x6x6=216 As 27 : 4ib, : : 216 : 321h. Ans.

3. If a solid globe of silver, of inches diameter, be worth 150 dollars; what is the value of another globe of silver, whose diameter is six inches ?

3X3X3=27 6x6x6+216 AS 27 : 150 :: 216 : $1200. Ans.

The side of a cube being given, to find the side of that cube wich shall be double, triple, &c. in quantity to the given cube.


Cube your y ven sidle, and multiply by the given proQube root of the product and required cube, audl the

4. If a cube of silver, whose side is two inches, bc worth 20 dollars ; I demand the side of a cube of like silver', whese value shall be 8 times as much ?

2x2x2=8 and 8X8=643/64=4 inches. Ans. 5. There is a cubical vessel, whose gide is 4 feet; } demand the side of another cubical vessel, which shall contain 4 times as much?

4x4X4=64 and 64x4=256 256x6,349+ft. Ans. 6. A cooper having a cask 40 inches long, and $2 in. chies at the bung diameter, is ordered to make another cask of the same shape, but to hold just twice as much; "what will be the bung diameter and length of the new çask ? 40x40x40x2=128000 then 128000=50,5+ length. 32X32X32X=65536 and 365536=20,3+-bung dian.

A General Hule for Extracting the Roots of all Powers.


1. Prepare the given number for extraction, by point ing off from the unit's place, as the requirei! root directs

2. Find the first figure of the root by trial, and subtract its

power froin the left hand period of the given number.

s. To the remainder bring down the first figure in the nert period!, and call it the dividend.

4. Involve theroit to the next inferior power to that which is given, and multiply it by the number klenoting the given power, for a divisor.

5. Find how many times the divisor may be liad in the dividend, and the quoticrt will be another figure of che root.

6. Involve the whole root to thicouver power, and subtract it (always) from as many periods of the given num ber as you have found figures 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 divi. sor, as before, and in like maglier proceed tili the whole be

Note.-When the number to be subtracted is greater than those periods from which it is to be taken, the last quotient figure must be taken less, &c.


1. Required the cube root of 135796,744 by the above general method.

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