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PROB. VI. The sum of any two numbers, and their products being given, to find each number.

RULE.

From the square of their sum, subtract 4 times their product, and extract the square root of the remainder, which will be the difference of the two numbers; then half the said difference added to half the sum, gives the greater of the two numbers, and the said half difference subtracted from the half sun, gives the lesser number.

EXAMPLES.

The sum of two numbers is 48, and their product is 442; what are those two numbers ?

The sum of the numb. 43×48=1849 square of do. The product of do. 442x 4-1768 4 times the pro.

Then to the sum of 21,5

[numb.

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✔81-9 diff. of the

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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 its square, shall produce the given number.

RULE.

1. Separate the given number inte periods of three figures each, by putting a point over the unit figure, and every third figure from the place of units to the left, and if there be decimals, to the right.

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

3. Subtract the cube thus found, from the said period, and to the remainder bring down the next period, calling this the dividend.

4. Multiply the square of the quotient by 300, calling the divisur.

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

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

NOTE.-If the subtrahend (found by the foregoing rule) happens to be greater than the dividend, and consequently cannot be subtracted therefrom, you must make the last quotient figure one less; with which find a new subtrahend, (by the rule foregoing) and so on until you can subtract the subtrahend from the dividend.

EXAMPLES.

1. Required the cube root of 18599,744.

18399,744(26,4 Root. Ans.

8

2x2=4×300-1200) 10399 first dividend.

7200

6x6=36×2=72×30=2160

6×6×6= 216

9576 1st subtrahend.

811200

26×26=676×300=203800)823744 2d dividend.

4x4-16x26=416×30= 12480

64

4X4X4=

825744 2d subtrahend.

NOTE.-The foregoing example gives a perfect root; and if, when all the periods e 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 cube root of 205379?

s. Of

4. Of

5. Of

6. Of

7. Of

8. Of

9. Of

10. Of

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1. Find by tria!, 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 root, 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 had to a greater degree of exactness.

EXAMPLES.

Let it be required to extract the cube root of 2. Assume 1,3 as the root of the nearest cube; then1,5×1,3×1,3-2,197 supposed cube.

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As 6,394

:

6,197

: : 1.S : 1,2599 root,

which is true to the last place of decimals; but might by repeating the operation, be brought to a greater exactness. 2. What is the cube root of 584,277056 ?

Ane, 8,36,

Required the cube root of 729001101 ?

QUESTIONS,

Ans. 900,0004

Shewing the use of the Cube Root.

1. The statute bushel contains 2150,425 cubic or solid inches. I demand the side of a cubic box, which shall contain that quantity?

2150,425 12,907 inch. Ans. Note. The solid contents of similar figures are in proportion to each other, as the cubes of their similar sides or diameters.

2. If a bullet 3 inches diameter, weigh 4lb. what will a bullet of the same metal weigh, whose diameter is 6 inches ?

3x3x3=27 6×6×6=216 As 27: 4lb. : : 216: 32lb. Ans.

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

$

SX3X3=27 6×6×6=216 As 27: 150 :: 216: $1200. Ans.

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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.

RULE.

Cube your given side, and multiply by the given proportion between the given and required cube, and the cube root of the product will be the side sought.

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

2×2×2=8 and 8×8=643/64-4 inches. Ans.

5. There is a cubical vessel, whose side is 4 feet; I demand the side of another cubical vessel, which shall contain 4 times as much?

4x4x4-64 and 64x4-256 256-6,349+ft. Ans. 6. A cooper having a cask 40 inches long, and 32 in

ches at the bung diameter, is ordered to make another cask of the same shape, but to hold just tee as much; what will be the bung diameter and length of the new cask?

40X40X40X2=128000 then 128000-50,S+ length. 32×32×32×2=65536 and 365566=40,3+bung diam.

A General Rule for Extracting the Roots of all Powers.

RULE.

1. Prepare the given number for extraction, by pointing off from the unit's place, as the required root directs. 2. Find the first figure of the root by trial, and subtract its power from the left hand period of the given number. S. To the remainder bring down the first figure in the next period, and call it the dividend.

4. Involve the root to the next inferior 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 subtract it (always) from as many periods of the given number 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 divisor, as before, and in like manner proceed till the whole be finished.

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.

EXAMPLES.

1. Required the cube root of 155796,744 by the above general method.

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