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Roots are sometimes denoted by a fractional index or exponent, of which the numerator indicates the power, or the number of times the number is to be taken as a factor, and the denominator indicates the root, or the number of equal factors into which that product is to be divided. Thus the square or second root of 12 is denoted by 124, the fourth root of by (3)+, and the square of the cube root of 27, or the cube root of the square of 27, is denoted by 273.

520. All the rational roots of whole numbers are also whole numbers, since every power of a fractional number is also a fractional number.

521. Prime numbers have no rational roots.

A composite number, to have a given rational root, must have the exponent of the power of each of its prime factors exactly divisible by the exponent of that root.

Note. — The number of composite numbers that have rational roots is comparatively small. The number of rational square roots of whole numbers from 1 to 250000 inclusive is only 500, and the number of rational cube roots of whole numbers from 1 to 8000000 inclusive is only 200.

522. The roots represented by the first ten numbers and their first six corresponding powers are shown in the following

TABLE.

1st Power, 2d Power, 3d Power, 4th Power, 5th Power, 6th Power,

1 2 3 4 5

6

8
9

10 1 4 9 16 25 36 49 64 811 100 1 8 27 64 125 216 343 512 729 1000 1 16 81 256 625 1296 2401 4096 6561 10000 1 32 243 1024 3125 7776 16807 32768 59049 100000 1 64 729 4096 15625 46656 117649 262144 531441 1000000

NOTE. — It will be observed by the table, that a rational square root can only be obtained from numbers ending in 1, 4, 5, 6, or 9; or in an even number of ciphers, preceded by one of these figures. It is true, also, that, if the square number ends in 1, its square root ends in 1 or 9; if in 4, its square root ends in 2 or 8; if in 9, its square root ends in 3 or 7; if in 6, its square root ends in 4 or 6; and if in 5, its square root ends in 5.

A perfect cube, however, may end in either of the nine digits, and in ciphers if the number of them is three or any multiple of three; also if the cube number ends in 1, its cube root will end in 1; if in 2, its cube root ends in 8; if in 3, its cube root ends in 7; if in 4, its cube root ends in 4; if in 5, its cube root ends in 5; if in 6, its cube root ends in 6; if in 7, its cube root ends in 3; if in 8, its cube root ends in 2; and if in 9, its cube root ends in 9.

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the square

523. The extraction of the square root of a number is the process of finding one of its two equal factors ; or of finding such a factor as, when multiplied by itself, will produce the given number.

524. The method generally adopted for extracting the square root depends upon the following principles :

1. The square of any number has, at most, only twice as many figures as its root, and, at least, only one less than twice as many. For the square of any number of a single figure consists of either one or two places of figures, as 12

: 1, and 92 = 81;

of

any number of two figures consists of either three or four places, as 10% = 100, and 992 = 9801; and the same law holds in regard to numbers of three or more figures. Therefore, when the square number consists of one or two figures, its root will consist of one figure; wheň of three or four figures, its root will consist of two figures; when of five or six figures, its root will consist of three figures; and so on. Hence, if a number be separated into as many periods as possible of two figures each, commencing at the right, to these periods respectively will correspond the units, tens, hundreds, &c. of the square root of the number.

2. The square of a number consisting of TENs and units is equal to the square of the tens, plus twice the product of the tens into the units, plus the square of the units. Thus, if the tens of a number be denoted by a and the units by b, the square of the number will be denoted by (a + b)2 a’ + 2 a b + b2. Then, by this formula, if a = 3, and 6 6, we have 3 tens + 6 units

30 + 6

36; and 362

(30 + 6)? 30+ 2 (30 X 6) + 62 = 1296. Or, analytically, a+b = 30 + 6

30+ 6

36 ato 30 + 6

30+- 6 (a+b)xa= 302+30X6

900+180 1080 (a+b)xb= 30X6+62

180+36 216 (a+b) 302+2X (30X6)+69= 900+360+36 1296

36

It is evident, as evolution is the reverse of involution (Art. 517), that from the process now given of obtaining a square may be deduced a method of extracting its root. Since the square of (a+b) is a’ + 2 ab + b?, the square root of a’ + 2ab + b2 must be a + b. Now it will be observed that a, the first term of the root, is the square root of a’, the first term of the square ; and if a’ be subtracted, there will remain 2 ab + b}, from which b, the second term of the root, is to be obtained. But 2 ab + b? is the same as (2 a + b) x b, therefore the remainder equals (2 a + b) X b. But as b, the units, is always much less than 2 a, twice the tens, we consider that 2 a X b is about equal to the whole remainder, and taking 2 a (which we know) as the trial divisor, we obtain b, the units. But as the true divisor is 2 a + b, we add the units to twice the tens and multiply the sum by the units, which gives a product equal to the whole remainder, or 2 ab + b?. Since

every number of more than one figure may be considered as composed of tens and units, we may have tens and units of units, tens and units of tens, tens and units of hundreds, &c. Hence, the principle just explained applies equally whether the root contains two or more than two figures.

525. To extract the square or second root of numbers.

OPERATION

Ex. 1. What is the square root of 1296 ? Ans. 36.

Beginning at the right, we separate the 1 2 9 6 3 6

number into periods of two figures each, by

placing a point () over the right-hand figure 9

of each period. Since the number of periods 66 396

is two, the root will consist of two figures, 3 9 6

tens and units. Then 1296 the square of

the tens plus twice the product of the tens 0

into the units, plus the square of the units.

The square of tens is hundreds, and must therefore be found in the hundreds of the number. The greatest number of tens whose square does not exceed 12 hundreds is 3, which we write as the tens figure of the root. We subtract the 9 hundreds, the square of the 3 tens, from the 12 hundreds, and there remain 3 hundreds; after which we write the figures of the next period, and the remainder is 396 twice the product of the tens into the units plus the square of the units. We have then next to find a number which, added to twice the 3 tens of the root, and multiplied into their sum, shall equal 396. By dividing this remainder

OPERATION

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by twice the three tens of the root, we may obtain the units, a num-
ber somewhat too large. But though it may be too large, it cannot be
too small, since the remainder 396 contains twice the product of the
tens into the units, and also the square of the units. We there-
fore make twice the three tens of the root 6 tens, a trial divisor,
with which we divide the 39 tens, exclusive of the 6 units, which
cannot form any part of the product of the tens by the units. The
quotient figure obtained, 6, must be the units figure of the root, or
a number somewhat larger. To determine whether it expresses the
real number of the units in the root, we annex it to the 6 tens, and
multiply the number 66, thus formed, by it. The product is 396,
which being subtracted, there is no remainder. Therefore 1296 is a
perfect square, and 36 the root sought.
2. What is the square root of 278784 ?

Ans. 528.
Since there are three

pe-
2 1 8 18 4 5 28 riods, the root will contain
2 5

three figures; the first two

may be considered as tens 10 2 2 87

and units of TENS. As the 2 0 4

square of tens cannot give 1048 8 3 8 4

less than hundreds, we must

find that square in the two 8 384

left-hand periods; and as we 0

have tens and units of tens,

the
square

of
528 x 528 278784 the tens plus twice the tens

into the units, plus the square of the units 2787 (nearly). We proceed then with the first two periods exactly the same as when the root consists of but two figures, and thus take from the given number the square of the 52 tens, which leaves a remainder of 8384. We now consider the given number 278784, as the square of a number consisting of 52 tens and a certain number of units, which square will of course equal the square of the tens plus twice the tens into the units, plus the square of the units. But the square of the tens, or (52) has already been taken from the given number, leaving a remainder, 8384, which must equal twice the tens into the units plus the square of the units. From this we readily obtain the units, just as when we had but two figures in the root.

RULE. — Separate the given number into as many periods as possible of figures each, by placing a point over the place of units, another over the place of hundreds, and so on.

Find the greatest square in the left-hand period; write the root of it at the right of the given number after the manner of a quotient in division, and subtract the second power from the left-hand period.

Bring down the next period to the right of the remainder for a dividend, and double the root already found for a trial divisor. Find how often this divisor is contained in the dividend, exclusive of the righthand figure, and write the quotient as the next figure of the root.

their square

PROOF

Annex the last root figure to the trial divisor for the true dirisor, which multiply by the last root figure and subtract the product from the dividend. To the remainder bring down the next period for a new dividend.

Double the root already found for a new trial divisor, and continue the operation as before, till all the periods have been brought down.

NOTE 1. When the product of any trial divisor exceeds its corresponding dividend, the last root figure must be made less.

If a dividend does not contain its corresponding divisor, a cipher must be placed in the root, and also at the right of the divisor; then, after bringing down the next period, this last divisor must be used as the divisor of the new dividend.

NOTE 2. - When there is a remainder after extracting the root of a number, periods of ciphers may be annexed, and the figures of the root thus obtained will be decimals.

NOTE 3. — If the given number is a decimal, or a whole number and a decimal, the root is extracted in the same manner as in whole numbers, except, in pointing off the decimals, either alone or in connection with the whole number, we place a point over every second figure toward the right, from the separatrix, filling the last period, if incomplete, with a cipher. The number of decimal places in the root will always equal the number of periods of decimals in the power.

NOTE 4.- If the given number is a common fraction, reduce it to its simplest form, if it is not so already, and extract the root of both terms, if they are perfect powers; otherwise, either find their product, extract its root, and divide the result by the denominator, or reduce the fraction to a decimal, and extract the root of the decimal.

NOTE 5. - When the given number is a mixed number, it may be changed to the form of a common fraction, or the fractional part may be reduced to a decimal, before attempting to extract the root.

EXAMPLES.

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3. What is the square root of 15445?

Ans. Wਤੇ 1 ਤੋਂ 5 = 4. What is the square root of go ?

Ans. .1936+:

V240 3 X 80 = 240;

.1936+

80 Or y = .0375; V.0375 = .1936+, 5. What is the square root of 3444736 ? Ans. 1856. 6. What is the square root of 998001 ?

Ans. 999. 7. What is the square root of Bong? 8. Extract the square root of 234.09 ?

Ans. 15.3. 9. What is the square root of 421?

Ans. 64

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