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%. 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 ?


36 23,8 2345 6031 13,57 + 98,555

,675+ ,054 6,708+


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 nume. rator, and the root of the denominator, for a new denomi nator.

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

EXAMPLES. 1. What is the square root of 10? Answers & 2. What is the square root of a

H 3. What is the square root of it? 4. What is the square root of 201? 5. What is the square root of 2481} ?

154 SURDS. 6. What is the square root of 1?

9128+ 7. What is the square root of ?

,7745+ 8. Required the square root of 361 ?

6,0207 +


ROOT PROBLEM I. A certain General has an army of 5184 men ;


many must be place in rank and file, to form them into a square ?

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Extract the square root of the given number.

5184-79 Anar Prob. II. A certain square pavement contains 20736 square stones, all of the same size; I demand how many are contained in one of its sides 20756x144 Ans.

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

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

EXAMPLES What is the mean proportional between 18 and 788

72x181296, and 1296-36 Ans. PROB. IV. To form any body of soldiers so that they may be double, triple, &c. as many in rank as in file.

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

LXAMPLES Let 13122 men be so formed, as that the number in Nnk may be double the number in file.

13122+2=6561, and 6561-81 in file, and 81 X9 3162 in rank.

PROB. V. Admit 10 hhds. of water are discharged through a leaden pipe of 24 inches in diameter, in a certain time; I deniand what the diameter of another pipe must be, to discharge fuur tnies u much water is the mainc time.

RUI.K. Square the giver diameter, and multiply said square by thie given proportion, and the square root of the preduct is the answer. 21-2,5, and 2,3x2,36,45 square.

4 given proportion 0,00 inoh. din. fine.

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 'suin, gives the tesser number.

The sum of two numbers is 48, and their product is 448; what are those two numbers ?

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

numb. +and


✓81.9 cit. of the



41 the ; diff.

Greatest number,
Least number,



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 cabe in the left hand period, and place its root in the quotient.

3. Subtract the cube thus found, from the said perib, 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 divi. 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 30, and place the product under the last; tnen 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.


1. Required the cube root of 18399,744.

18399,744(26,4 Root. Ans.

( 8

2x2=4x300=1200) 10399 first dividend.

7200 6x6=36X2=72X30=2160

6x6x6= 216

9576 1st subtrahend. 26 x 26-676 x 300=203800) 823744 2d dividend.

811200 4X4=16x26=416x30= 12480

4X4X4= 64

825744 2d subtrahend.

( 52,7

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

Ansuers. 2. What is the cube root of 205379 ?

59 3. Of

614125 ?

85 4. Of


346 3. Of

146363,183? 6. Of

29,503629 ?

3,09 7. Of

80,763 ?

4,32+ 8. Of


,546 9. Ot

,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 supo posed 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.

Let it be required to extract the cube root of %.

Assume 1,3 as the root of the nearest cube ; then 1,5X1,5X1,3=2,197=supposed cube. Then, 2,197 2,000 given number. 2


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As 6,394 6,197

1,3 1,2599 ront, 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?

ne 8,96

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