THE FORMATION OF POWERS, AND EXTRACTION OF ROOTS.

We call the power of a number, the products which we find by multiplying it by itself a certain number of times; and we call the root of the power, the number, which, being multiplied by itself a certain number of times, produces this power.

After this definition, we see very well, there is no difficulty in finding the power required of any number whatever.

Every number is itself its first power; the 2nd. power, or the square of a number, is the product of that number, multiplied by itself; the 3d power, or the cube of a number, is the product of that number by its square, the 4th power of a number is &c.

&c.

We must remark, that all the powers of 1 are 1; for 1 multiplied by itself, as often as we please, can give but 1. If there is no difficulty in finding any power whatever of a number, there is none, also, in obtain ing the root of a number, considered as a power, we call the square root, the root of the second power, the cube root, that of the third, the fourth root that of &c. and so on.

We shall here only lay down the method of extracting the square and cube roots, as being the only ones of which, we may have any need in Arithmetic.

EXTRACTION OF THE SQUARE ROOT.

The method employed to extract the square root of a number, is founded on the knowledge we have of

the formation of a square. But we must first observe. that every square which is composed but of two figures, can have only one figure for its root, and then, by means of the multiplication table, it will always be easy to know the square root of a number composed of two figures, or of the greatest square which shall be therein contained. Thus, the question is reduced, to find the root of a square, composed of more than two figures.

I say, first, that this square must have tens and units for its root; for 100 which is the least number of three figures, has for its root 10, which is composed of two figures. Let us see then, what a square is composed of, which has tens and units for its

root.

Let us take 24 for the root, and form of it the square, taking care to write separately each of the partial products.

I have then 576 for the square of 24; I see that this number is composed of 400, which is the square of the tens, and of twice 80 or 160, which is the double of the tens by the units; and lastly of the square of the units. With this knowledge, we can extract the square root of any number whatever.

24

24

16

80

80

400

576

Let it be proposed, for example, to extract the root of

I remark at first, that this number, being composed of more than two figures, has necessarily tens and units for its root; then it is composed of the square of the tens, of the double of the tens by the units, and of the square of the units; but the square of the tens being a number of hundreds, has two places to its right; if, then, we separate the two

first figures of this number towards the right, it is certain that the part which is on the left, will contain the square of the tens.

But I remark, that this number, being itself composed of more than two figures, has still tens and units for its root; thus, also, it contains the square of the tens, the double product of the tens by the units, and the square of the units, but the square of the tens is a number of hundreds, which has two places at its right; if then I separate the two first figures to the right of this number; the part on the left will contain the square of the tens; but this number, being itself still composed of more than two figures, contains tens and units for its root, therefore we must still separate two

In a number, any figure whatever, which exprefsed tens in comparison to the figure which is on its right, may be confidered as a number of units in comparison to the figure which is on its left.

figures on the right; then there remains only one figure, which must contain the square of the tens.

In effect, this square is contained therein, but with an excess which results from the double product of the tens by the units. I then take the root of the greatest square comprised in 6, this greatest square is 4, and its root 2, which I write, as we see, to the left of the number, as if it were a divisor; now I square the tens of the root, which gives me 4, which I subtract from 6, and to the right of the remainder 2, I bring down the next period.

I remark, that the number 220, from which I subtract the square of the tens, contains no more than the double product of the tens by the units, and the square of the units; but the double product of the tens by the units, is a number of tens, which has one place to the right; then if I separate one figure to the right of this number, the part on the left will contain the double product of the tens by the units; if, therefore, I divide this number by the double of the tens, I shall have the units in the quotient; the double of the tens is 4, which I write below the root, and divide 22 by 4; the quotient is 5, but I try it before I write it, and this is the method.

Since 220 is composed of the double product of the tens by the units, and of the square of the units, if I take the square of 5, which I suppose to be the units, and add it to the product of 4 by 5, which will be the double of the tens by the units, I shall have a sum, which ought to admit of being subtracted from 220; I see that the subtraction is not possible; thence I conclude, that 5 is too great; I try 4, which is found to be the right figure, I then write this figure beside the tens of the root, and I also write it at the

right hand of the number which is below; then, looking upon 44 as the divisor, and 4 as the quotient, I perform the multiplication, and operate as in division ; the subtraction being made, there remains 44.

Beside this remainder I write 54, which is the next period; then reasoning as above, I say, that 4454 contains only the double product of the tens by the units, and the square of the units; but the double product of the tens by the units, is a number of tens which has one place to its right; if then I separate one figure to the right of this number, the part 445 which remains to the left, ought to contain the double product of the tens by the units; I therefore double the tens which are now 24, which gives me 48, which I write below the root, after having barred the number 44 which served me before.

I divide 445 by 48, and I find for the, quotient 9, which I write after the tens of the root, and beside the double of the tens by the units; I proceed as above, and have 53 for remainder; I bring down the last period, I separate the first figure to the right, I double the figures of the roots, to obtain a new divisor, which gives me 1 in the quotient, and 323 for remainder; so that the root of the greatest square contained in the number proposed, is 2491.

For the proof; the square of the root, joined to the remainder, must be equal to the given number.

it

The number 2491 is the square root of the number proposed within a unit, but if we wished to have more exactly within a tenth, a hundredth, or &c. there would be no difficulty; we might continue the operation, after having added to the number twice as many cyphers as we wish to have decimals in the root; this manner of proceeding is founded upon