A ROOT TAKES ITS NAME or number, FROM THE NUMBER OF TIMES IT IS CONTAINED AS A FACTOR, IN THE CORRESPONDING POWER.

Hence also,

The words POWER and ROOT are correlative terms. That is, if one number be a certain power of another; the latter is the same root of the foriner. Thus, 16 is the fourth power of 2, and 2 is the fourth root of 16.

We have seen that when a root is given, there is no difficulty in involving it to any power. For Involution is performed by multiplication: and any two or more numbers, whatever, may be multiplied together, and the exact product obtained. But there are many numbers, whose roots cannot be exactly ascer tained. Thus, the square root of 2 cannot be exactly expressed. It is nearly 1.41421356237+

A Root, which cannot be exactly expressed, is called a SURD or IBRATIONAL number. On the other hand,

A Root which can be exactly ascertained, is called a RATIONAL number. A power of a rational number, is called a PERFECT power. Thus 9 is a perfect square, its root being 3; but it is an imperfect cube, its root being 2.0800865+ Also, the square root of 9 is a rational number, and its cube root a surd. Surds may be expressed in decimals, with a degree of accuracy, sufficient for all prac. tical purposes.

As there is a mode of indicating powers; so likewise there is an appropriate corresponding method of indicating roots. As in powers, a whole number, equal to the number of the power, is placed at the right; so in roots, a Fraction whose denominator is the number of the root, is employed in a similar manner. Thus is the index of the 2d. root, 4, of the 3d., and so on. The square root of 4 is then expressed 4; the 8th. root of 7, 74, &c. The numerator of the Fraction, in such cases, is always 1.

There is, likewise, another mode of indicating roots. The character, V, prefixed to a number denotes the square root, and may be employed, instead of the index. The same character is used to denote other roots, the number of the root being written over it. Thus,,,, denote the third, fourth and fifth roots, respectively, and are used, instead of the indices,, and .

The process of finding roots, is commonly called the EXTRACTION OF ROOTS; and, when a root is ascertained by an arithmetical operation, it is said to be

EXTRACTED.

one.

EXTRACTION OF THE SQUARE ROOT.

§ C. When two numbers are multiplied together, the product never contains more figures than both factors together, nor fewer than the same number, less For if 99, for example, (the greatest number, consisting of two figures,) be multiplied by 100, the product is 9,900. If, instead of 100, the multiplier had been 99 also, or 98, or any less number, the product would of course been less than 9,900. But 9,900 contains only four places. Hence, it is evident, that no two factors, consisting of two figures each, can produce a product, contain ing more than four figures; that is, more than there are in both factors. The same mode of proof will apply to any other case.

On the other hand, 10 and 10 are the smallest numbers, consisting each of two figures. But 10X10=100, which contains three figures, or one less than both factors. The same mode of proof will apply to other cases. Hence, no square can contain more figures than twice as many as its root contains, nor few. er than the same number, less one. Hence, when a square root is required, we can always determine how many figures it will contain. The most convenient mode of doing this, is to begin at the right of the given square, and point off the number into periods of two figures each. Thus, how many figures in the square root of 363729617 Put a point over units, and afterwards over each scc

ond figure, thus, 36372961. There are four points: therefore the root contains four figures. In extracting the square root, the first thing to be done is to point off the given number in this manner.

1. In a square court, are contained 576 square rods. How many rods in length is one side of the court?

We will tell you in this instance, at first, that the side of the square is 24. Now involve 24, and set down the product of every two figures separately, thus. 4 times 4 units are 16 units; 4 times 2 tens are 8 tens, or 80; 2 tens into 4 units are 8 tens, or 80, and 2 tens into 2 tens are 4 hundreds=400. The little square A represents the first product, the long figure B, the second, the long figure C, the third, and the large square D, the fourth. Let them all be added, and the square A B C D below, will represent the total product, 576. Now suppose we did not know the root. On pointing off as above directed, we should know that it consisted of two figures, tens and units, because there are two points. PROCESS.

4 rods.

20 rods.

4 rods.

20 rods.

24

24

16

80

80

400

576

20 rods.

4 rods.

16

80 B

D

400

20 rods.

4 rods.

Now the square of the tens' figure is hundreds, and will of course be the great. est square to be found in the left hand period. This is 400, and its root 20. This square (the figure D.) being taken away, 176 remain (=the figures A B, and C.) B and C are each 20 rods long, (being equal in length to D,) and are just as wide as A. All three may, therefore, be placed in a row, thus,

If this figure, ab cd, were divided by the length of the side a b, the quotient would be a c (the side of A,) that is, the units' figure of the root. Now we do no

know the length of A, but we know that B and C are each 20 rods long, and, of course, both together are 2X20-40 rods long. 40, then, will be sufficiently near to divide by. 176-40-4 (neglecting remainder.) . We suppose 4, then, to be the units' figure, or the side of A. This 4 we add to 40, to get the length of abcd, which is 44. Then, the length, 44X4 the width, 176, the size of the figure a b c d. This agrees with its actual size, and hence we know 4 to be the units' figure of the root. Therefore 24 rods is the length of one side of a court containing 576 square rods.

The above operations show the process here illustrated. The large square, 400, is first subtracted; then the remainder is divided by 40, which is twice 20, the root already found. The quotient found is 4. This is added to 40 making 44, and the sum multiplied by 4 produces 176, which, being subtracted, leaves no remainder.

In one of the operations the cyphers are retained at the right of the square number 400, and of its root 20. In the other, these cyphers are dropped, as in common division. When they are dropped, however, a cypher must evidently be understood at the right of the divisor.

The intelligent pupil will easily see that when the root has more figures, the process is similar. For the first two periods, it is exactly the same. The third period must then be brought down, the whole root already found doubled for a divisor, (understanding a cypher at the right, as above,) and so on as before. All that is to be observed in this case is, that each divisor must be obtained by doubling the whole root already found. This may be proved by a diagram, as above. Another set of figures like A, B and C must be constructed outside of those. If the pupil does not succeed in the demonstration, it is recommended to the teacher to exhibit it.

It is almost needless to remark that we may sometimes have a remainder after the last period is brought down. Thus, in the example above, if the num ber were 580, instead of 576, 24 would still be the nearest whole number root, and a remainder of 4 would be left. To such a remainder, we may annex peri. ods of cyphers, and continue the root to decimals. Each period, so annexed, must of course contain two cyphers. Likewise, if any dividend is too small to contain the divisor, we must put a cypher in the root, and bring down another period.

2. Extract the square root of 5,499,

025.

5499025(2345 ROOT.

4 43 149

3:129

464/2090

4/1856

4685/23425

3. Find the square root of 2.

2(1.4142+ ROOT.

1

24/100
4 96

281 400
1 281

2824/11900

411296

23425

00

In extracting the root of 2, we are obliged to annex periods of cyphers, after obtaining one figure in the root. Such a root as this, must, of course, be surd; for every dividend ends with a cypher, and the first figure of each subtrahend is the product of some figure by itself, since the last figure of every divisor is the same as the quotient figure by which it is multiplied. But no significant figure, multiplied by itself, produces a product ending with a cypher. Hence, there will always be a remainder, and the root will be, of course, infinite. If, then, there is a remainder, when all the significant figures of any number have been employed in Evolution, the root of that number is a surd. The same is true in case of other roots, as well as of the square.

It will be seen, that, as we carry a root to decimals, by annexing periods of cyphers below the units' place, so we should annex periods of significant decimals, if they were in the given number. Hence, decimals are to be pointed off by twos, from the units' place.

4. A farmer wished to lay out a field, in the form of a square, to contain 529 square rods; how long must he have made one side ? A. 23 rods. What is the length of one A. 31 ft. 26. Of 625.

5. A square floor contains 961 sq. ft. side?

A. 28.

Of 676. A.

6. Find the sq. root of 784. A. 25. Of 487,204. A. €98. Of 638,401. A. 779. Of 556,516. A. 746. Of 441. A. 21. Of 1,024. A. 32. Of 1,444. A. 38. Of 2,916. A. 54. Of 6,241. A. 79. Of 9,801. A. 99. Of 17,956. A. 134. Of 32,761. A. 181. Of 39,601. A. 199. Of 488,601. A. 699.

Of 83. A. 9.1104336.

7. Find the sq. root of 69. A. 8.3066239. Of 97. A. 9.8488578. Of 299. A. 17.2916165. Of 222. A. 14.8996644. Of 282. A. 16.7928556. Of 394. A. 19.8494332. Of 26.4386081

Of 699. A.

351. A. 18.7349940. 2889757. Of 989. A. 31.4483704. Of 999. A. 31.6069613. Of Of 687. A. 26.2106848.

397. A. 19.9248588.

8663690.

Of 979. A. 31..

Of 892. A. 29.

To find the root of a fraction, take the root both of numerator and denominator, or, if this cannot be done, reduce the fraction to a decimal, and extract its root. The same may be done with mixed numbers.

249001

8. What is the sq. root of 25? A. §. Of 160881? A. 481 Of 22718? A. 183. Of 439338?

499. 6162

480249

487

483025

A. 58. Of

695

Of 12
5 A.

9. Find the sq. root of 2. A. .8660254. .645497. Of 173. A. 4.168333. Of A. .193649167. Of. A. .83205. Of. A. .288617394+ From the above illustrations and examples, we have the rule,

1. HAVING POINTED OFF, SUBTRACT FROM THE HIGHEST PERIOD THE GREATEST SQUARE CONTAINED IN IT, PLACE THE ROOT IN THE QUOTIENT, AND TO THE REMAINDER BRING DOWN THE NEXT PERIOD FOR A DIVIDEND.

II. DOUBLE THE ROOT ALREADY FOUND, (UNDERSTANDING A CYPHER AT THE RIGHT,) FOR A DIVISOR, AND DIVIDE THE DIVIDEND BY IT, FOR THE NEXT FIGURE OF THE ROOT.

III. ANNEX THIS FIGURE TO THE DIVISOR, WHICH, SỌ INCREASED, MULTIPLY BY THE SAME FIGURE FOR A SUBTRAHEND. IV. SUBTRACT THE SUBTRAHEND FROM THE DIVIDEND, TO THE REMAINDER BRING DOWN THE NEXT PERIOD FOR A NEW DIVIDEND, AND SO PROCEED.

The proof is by Involution.

NOTE. The roots of many powers may be found by repeated extractions of the square root. Thus, the square root of the square root is the 4th root; the square root of the 4th root the 8th root, and so on. The same may be done by means of the cube root, and by the square and cube roots combined.

EXTRACTION OF THE CUBE ROOT.

§. CI. By similar reasoning to that in § C. it may be shown, that, when three numbers are multiplied together, the product can never consist of more figures than all the factors together, nor of fewer than the same number less two. Hence, we infer, in like manner, that if any number is pointed off into periods of three figures each, from the units' place, the number of periods will be equal to the number of figures in its cube root. Thus, how many figures in the cube root of 27054036008? Point, thus, 27054036008 There are four periods, and, of course, the root consists of four figures.

1. A cubic solid contains 13,824 cubic feet. How many feet in length is one edge of the solid?

13,824 is the cube of 24. One side of the solid, then, is 24 feet long. This number, 24, is the same as that used in ex. 1, of the last §. Let it be involved to the cube, the products of the digits composing it, being preserved distinct. In § C. we have already performed the first multiplication, and the several products were 400+80+80+16, represented by distinct diagrams. Multiply by 24 again, and each product may be represented by a solid.

400+80+80+16=576

Thus, (taking 4, the units' figure,) 16, (=the square, A, § C.)X4= 64-the cube A. 80 (-long figure B, C,) X4-320-the solid B. 80 (=figure C, §C,) ×4=320 the solid C. 400 (=square D.) X4 1,600 the solid D. Then, (taking the 2 tens,=20) 16×20= 320 the solid E. 80x20 1,600-the solid F. 80x20=1,600== the solid G. 80x400-8,000 the cube H.

N

Now, if the solids A, B, C, and D, should be placed immediately in front of E, F, G, and H, respectively, and the whole brought close together, a cube, I, K, L, M, N, O, would be formed 24313,824. In this cube, the solid H, 8,000, is the cube of the tens in the root; the solids, B, C, E,=320 each, are products of the tens into the square of the units; the solids, D, F, G,=1,600 each, are products of the units into the square of the tens; and the solid A, 64, is the cube of the units. Let the given number now be pointed off, thus,

L.

K