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NOTE. The root will contain just as many places of figures, as there are periods or points in the given power; and they will be integers or decimals respectively, as the periods are so, from which they are found, or to which they correspond; that is, there will be as many integral or decimal figures in the root, as there are periods of integers or decimals in the given number.

TO EXTRACT THE SQUARE ROOT.

RULE.*

1. Having distinguished the given number into periods, find a square number by the table or trial, either equal to, or next less than the first period, and put the root of it on the right of the given number, in the manner of a quotient figure in division, and it will be the first figure of the root required.

In order to show the reason of the rule, it will be proper to premise the following

LEMMA. The product of any two numbers can have at most but as many places of figures, as are in both the factors, and at

least but one less.

DEMONSTRATION. Take two numbers, consisting of any number of places, but let them be the least possible of those places, namely, unity with cyphers, as 1000 and 100; then their product will be 1 with as many cyphers annexed, as are in both the numbers, namely, 100000; but 100000 has one place less than 1000 and 100 together have; and since 1000 and 100 were taken the least possible, the product of any other two numbers, of the same number of places, will be greater than 100000; con- . sequently the product of any two numbers can have at least but one place less than both the factors.

Again, take two numbers of any number of places, that shall be the greatest of these places possible, as 999 and 99. Now 999×99 is less than 999 × 100; but 999 × 100 (=99900) contains only as many places of figures, as are in 999 and 99; therefore 999×99, or the product of any other two numbers, consisting of

2. Subtract the assumed square from the first period, and to the remainder bring down the next period for a dividend.

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

the same number of places, cannot have more places of figures than are in both its factors.

COROLLARY 1. A square number cannot have more places of figures than double the places of the root, and at least but one less.

COR. 2. A cube number cannot have more places of figures than triple the places of the root, and at least but two less. The truth of the rule may be shown algebraically thus : Let N the number, whose square root is to be found. Now it appears from the lemma, that there will be always as many places of figures in the root, as there are points or periods in the given number, and therefore the figures of those places may be represented by letters.

Suppose to consist of two periods, and let the figures in the root be represented by a and b.

Then a+ba2+2ab+b2=N= given number; and to find the root of Nis the same, as finding the root of a2+2ab+b2, the method of doing which is as follows:

1st divisor a)a2+2ab+b2(a+b= root.

a2

2d divisor 2a+b)2ab+b2

2ab+b2

Again suppose to consist of 3 periods, and let the figures of the root be represented by a, b, and c.

Then

2

a+b+c=a2+2ab+b2+2ac+2bc+c2, and the manner

of finding a, b, and c will be, as before thus,

1st divisor a)a2+2ab+b2+2ac+2bc+c2 (a+b+c=root.

a2

2d divisor 2a+6)2ab+b2

2ab+b2

4. Consider what figure must be annexed to the divisor, so that if the result be multiplied by it, the product may be equal to, or next less than the dividend, and it will be the second figure of the root.

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

6. Find a divisor as before, by doubling the figures already in the root; and from these find the next figure of the root, as in the last article; and so on through all the periods to the last.

NOTE 1. When the root is to be extracted to a great number of places, the work may be much abbreviated thus: having proceeded in the extraction by the common method till you have found one more than half the required number of figures in the root, the rest may be found by dividing the last remainder by its corresponding divisor, annexing a cypher to every dividual, as in division of decimals; or rather, without annexing cyphers, by omitting continually the first figure of the divisor on the right, after the manner of contraction in division of decimals.

NOTE 2. By means of the square root we readily find the fourth root, or the eighth root, or the sixteenth root, &c. that is, the root of any power, whose index is some power of the number 2; namely by extracting so often the square root, as is denoted by that power of 2; that is, twice for the fourth root, thrice for the eighth root, and so on.

3d divisor 2a+2b+c)2ac+2bc+c2

2ac+2bc+c2

Now the operation in each of these cases exactly agrees with the rule, and the same will be found to be true, when N consists of any number of periods whatever.

TO EXTRACT THE SQUARE ROOT OF A
VULGAR FRACTION.

RULE.

First prepare all vulgar fractions by reducing them to their least terms, both for this and all other roots. Then

1. Take the root of the numerator and that of the denominator for the respective terms of the root required. And this is the best way, if the denominator be a complete power. But if not, then

2. Multiply the numerator and denominator together; take the root of the product: this root, being made the numerator to the denominator of the given fraction, or the denominator to the numerator of it, will form the fractional root repuired.

Vab a

That is, ab

b

=

b

And this rule will serve, whether the root be finite or infi

nite.

Or 3. Reduce the vulgar fraction to a decimal, and extract its root.

EXAMPLES.

1. Required the square root of 5499025.

5499025(2345 the root.

43/149
3129

464/2090
4/1856

4685123425
23425

2. Required the square root of 184 2.

184′2000(13'57 the root.

1

23/84

369

26511520
51325

2707/19500
18949

551 remainder

3. Required the square root of 2 to 12 places, 2(1'41421356237 + root

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