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5. What is the 5th power of 195? Ans. 281950621875.

6. What is the 6th power of §? Ans. ffa.

7. Required the 2d power of 4698.

8. Required the 2d power of 6031. Ans. 36372961.

9. What is the 13th power of 7? Ans. 96889010407.

10. What is the 12th power of 6?

11. What is the loth power of 9? Ans. 205891132094649.

12. What is the 4th power of 4.367? Ans. 363.691179+.

13. Involve the following numbers to the powers denoted by their respective exponents: (2£)5, 1.04'5, and (3f)\

Ans. 157-&VV; 1.800943+; 116*£§f

EVOLUTION.

517. Evolution, or the extraction of roots, is the process of finding the roots of quantities. It is the reverse of involution.

518. The root of a quantity or number is such a factor as, being multiplied into itself a certain number of times, will produce that quantity or number.

The root takes the name of the power of which it is the correlative term. Thus, if the number is a second power, the root is called the second or square root; if it is the third power, the root is called the third or cube root; if it is the fourth power, its root is called the fourth or biquadrate root; and so on.

Rational roots are such as can be exactly obtained.
Surd roots are such as cannot be exactly obtained.

519. Roots are usually denoted by writing the radical sign, \/, before the power, with the index of the root over it; in case, however, of the second or square root, the index 2 is omitted. Thus, the third root of 27 is denoted by 4/27, the second root of 16 is denoted by and the fourth root of ^ is denoted by .v/£.

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 12*, the fourth root of § by (f and the square of the cube root of 27, or the cube root of the square of 27, is denoted by 27*.

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 i 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.

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

[merged small][table]

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, i f 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 1n 3; if in 8, its cube root ends in 2; and if in 9, its cube root ends in 9.

EXTRACTION OF THE SQUARE ROOT.

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 Is = 1, and 9s = 81 ; the square of any number of two figures consists of either three or four places, as 10a = 100, and <J9a = 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; when 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)1 o? + 2 a b + J2. Then, by this formula, if a = 3, and 6=6, we have 3 tens + 6 units = 30 + 6 = 3(3; and

362 = (30 + 6)a = 302 + 2 X (30 X6) + 6' = 1296. Or, analytically,

a + & = 30 + 6 = 30+ 6 =36

a + 6 = 30 + 6 = 80+ 6 =36

(a + 6)Xa = 302+30X6 = 900+180 = 1080

(a + ft)X6 = 30X 6 +62 = 180+36 = 216

(a + 6)a = 30+-2X(30X6)+C2 = 900+360+36 = 129S

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 -f- b) is a2 -)- 2 a b -\- b2, the square root of a2 -f- 2 a b -\- b3 must be a -j- 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 a2 be subtracted, there will remain 2 a b -)- b*, from which b, the second term of the root, is to be obtained. But 2 a b -\- h* is the same as (2 a -f- 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 -f- 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 a b -f- 52.

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.

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

Operation. Beginning at the right, we separate the

1 9 Q R qc number into periods of two figures each, by i J oo placing a point (•) over the right-hand figure "of each period. Since the number of periods

6 6 3 9 6 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 hundred*; 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 by twice the three tens of the root, we may obtain the units, a number 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 therefore 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 0 tens, and multiply the number 66, thus formed, by it. The product is 3'JC, 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.

operatiou. Since there are three pe

278784-1528 riods, the root will contain

2 5 three figures; the first two

may be considered as. tens

1 0 2 2 8 7 and units of Tens. As the

2 0 4 square of tens cannot give

1 0 4 8 8 3 8 4 'ess than nun('re<Js, we must

g g o ^ find that square in the two

left-hand periods; and as we

0 have tens and units of tens,

moor. their square = the square of

528 X 528 = 278784 !he tu"s I,!U9 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)a has already been token 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 man;/ periods as possible of two 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 it contained in the dividend, exclusive of the righthand figure, and write the quotient as the next figure of the root

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