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2. What is the square root of 1296 ? 3. Of

56644 ? 4. Of

5499025 ? 5. Of

36372961 ? 6. Of

184,2 ? 7. Of

9712,693809 ? 8. Of

0,45369 ? 9. Of

,002916 ? 10. Of

45

Answers.

36 23,8 2345 6031 13,57 + 98,553

,673 + ,054 6,708+

TO EXTRACT THE SQUARE ROOT OF
VULGAR FRACTIONS.

RULE. Reduce the fraction to its lowest terms for this and all other roots ; then

1. Extract the root of the numerator for the new numerator, and the root of the denominator, for a new denoma nator.

2. If the fraction be a surd, reduce it to a deciinal, and extract its root.

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APPLICATION AND USE OF THE SQUARE

ROOT Problem I. A certain General has an army of 5184 men ; how many must he place in rank and file, to form them into a square?

RULE.
Extract the square root of the given number.

✓ 5184=72 Ans. PROB. II. A certain square pavement contains 20755 square stones, all of the same size ; 1 demand how many are contained in one of its sides ? ✓ 20736=144 Ans.

PROB. III. To find a mean proportional between two Qumbers.

RULE. Multiply the given numbers together, and extract the square root of the product.

EXAMPLES.

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What is the mean proportional between 18 and 72 ?

72x18=1290, and ✓1296336 Ans. Prob. IV. To form any body of solliers so that they may be double, triplc, &c. as many in rank as in file.

RULE. Extract the square root of 1-2, 1-3, &c. of the given number of men, and that will be the number of men in file, which double, triple, &c. and the product will be the nuinber in rank.

EXAMPLES.

Let 13122 men be so formed, as that the number in rank

may be double the number in file. 13122-2=6561, and 6561=81 in file, and 81 X2 =162 in rank.

Prob. V. Admit 10 hhde. of water are discharged through a leaden pipe of 21 inches in diameter, in a certain tine; 1 demand what the diameter of another pipe must be, to discharge four times as much water in the same time.

RULE. Square the giver diameter, and multiply said square by the given proportion, and the syuare root of the proHuct is the answer. 21=2,5, and 2,5x2,5=6,25 square.

4 given proportion.

25,00=5 inch. diam. Ans.

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 lialf sum, gives the lesser number.

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

The sum of the viumb. 43x43=1849 square of do. The product of do. 442x 4=1768 4 times the pro. Then to the $ sum of 21,5

numb. tand

4,5

✓81=9 citro of the

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EXTRACTION OF THE CUBE KOOT. 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 into periods of three figures each, by putting a point over the init figure, and every third figure from the place of units to the left, and if there le 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 it the divisor.

5. Seek how often the divisor may be liad in the divis dend, 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 S0, and place the product under the last; tren under these two products place the cube of the last quotient figure, and add them together, calling their sum the subtrahend.

7. Subtract the subtrabend 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 finishgl.

Note.--If the subtrahend (found by the foregoing rule) happens to be greater than the dividend, and consequently camot be subtracted therefroin, you must nake 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,

EXAMPLE'S.

1. Required the cube root of 18399,744.

18399,744(26,4 Root. Ans.

. 8

2x24x300=1200) 10399 first dividend.

7200 6x6=36X2=72X30=2160

6x6x6= 216

9576 1st subtrahend. 26 x 26=676x300=203800)823744 2d dividend.

811200 4X4=16x26=416x30= 12480

4x4X4=

64

328714 2d subtrahend.

,546

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 cube root of 205379 ?

59 3. Of

614125 ?

85 4. Of

41421736?

346 5. Of

146863,183 ?

52,7 6. Of

29,503629 ?

3,09 7. Of

80,763 ?

4,32-+ 8. Of

,162771336 ? 9. Of

,000684134

,088+ 10. Of 122615327232?

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 root, or an approximation to it.

S. 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 ; then 1,3x1,3x1,3=2,197 = supposed cube. Then, 2,197 2,000 given number. 2

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 a greater exactness. 2. What is the cube root of 584,277056 ?

Ano. 8,36.

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