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What is the square of 17,1 ?
What is the square of ,085 ?
What is the cube of 25,4?
What is the biquadrate of 12?
What is the square of 74?

Ans. 292,41

Ans. 16387,064
Ans. 20736

Ans. 52%

EVOLUTION, OR EXTRACTION OF ROOTS. WHEN the root of any power is required, the business of finding it is called the Extraction of the Root. The root is that number, which by a continual multiplication into itself, produces the given power.

Although there is no number but what will produce a perfect power by involution, yet there are many numbers of which precise roots can never be determined. But, by the help of decimals, we can approximate towards the root to any assigned degree of exactness.

The roots which approximate, are called surd roots, and those which are perfectly accurate are called rational


A Table of the Squares and Cubes of the nine digits.


Squares. Cubes.

4|1|2| 3| 4| 5 |
1|49|16| 25

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216343 512 | 729

| 8|27|64 | 125

EXTRACTION OF THE SQUARE ROOT. Any number multiplied into itself produces a square. To extract the square root, is only to find a number, which being multiplied into itself, shall produce the given



1. Distinguish the given number into periods of two figures each, by putting a point over the place of units, another over the place of hundreds, and so on; and if there are decimals, point them in the same manner, from units towards the right hand; which points show the number of figures the root will consist of..

2. Find the greatest square number in the first, or left

hand period, place the root of it at the right hand of the given number, (after the manner of a quotient in division) for the first figure of the root, and the square number under the period, and subtract it therefrom, and to the remainder bring down the next period for a dividend.

5. Place the double of the root, already found, on the left hand of the dividend for a divisor.

4. Place such a figure at the right hand of the divisor, and also the same figure in the root, as when multiplied into the whole (increased divisor) the product shall be equal to, or the next less than the dividend, and it will be the second figure in the root.

5. Subtract the product from the dividend, and to the remainder join the next period for a new dividend.

6. Double the figures already found in the root, for a new divisor, and from these find the next figure in the root as last directed, and continue the operation in the same manner, tal you have brought down all the periods.

Or, to facilitate the foregoing Rule, when you have brought down a period, and med a dividend, in order to find a new figure in the root, you may divide said dividend, (omitting the right hand figure thereof,) by double the root already found, and the quotient will commonly be the figures sought, or being made less one or two, will generally give the next figure in the quotient.

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1. Required the square root of 141225,64. 141225,64(375,8 the root exactly without a remainder;


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but when the periods belonging to any given number are exhausted, and still leave a remainder, the operation may be continued at pleasure, by annexing periods of cyphers, &c..

0 remains.

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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


a new denomi

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

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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?


Extract the square root of the given number.

✓5184 72 Ans.

PROB. II. A certain square pavement contains 20736 square stones, all of the same size; I demand how are contained in one of its sides?


20736=144 Ans.

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


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


What is the mean proportional between 18 and 72 |, 72×18-1296, 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.


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 number in rank.


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

13122÷2-6561, and

162 in rank.

6561-81 in file, and 81x2

PROB. V. Admit 10 hhds. of water are discharged through a leaden pipe of 24 inches in diameter, in a cer tain time; I demand what the diameter of another pipe must be, to discharge four times as much water in the same time.


Square the given diameter, and multiply said square by the given proportion, and the square root of the product is the answer.

212,5, and 2,5×2,5-6,25 square.

4 given proportion.

25,005 inch. diam. Ans.


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


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 sum, gives the lesser number.


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

The sum of the numb. 43×43-1849 square of de. The product of do. 442× 4=1768 4 times the pro. Then to the sum of 21,5 [numb.

81-9 diff. of the



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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



1. Separate the given number into 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 it the divisor.

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