III. Take three times the square of the root just found for a divisor, and see how often it is contained in the dividend, and place the quotient for a second figure of the root. Then cube the number thus found, and if its cube be greater than the first two periods of the given number, diminish the last figure by 1; but if it be less, subtract it from the first two periods, and to the remainder bring down the first figure of the next period, for a new

dividend.

IV. Take three times the square of the whole root for a new divisor, and seek how often it is contained in the new dividend; the quotient will be the third figure of the root. Cube the number thus found, and subtract the result from the first three periods of the given number, and proceed in a similar way for all the periods.

If there is no remainder, the number is a perfect cube, and the root is exact: if there is a remainder, the number is an imperfect cube, and the root is exact to within less than 1.

Ans. 364.

Ans. 3002.

Ans. 78, with a remainder 8697.

91632508641 Ans. 4508, with a remainder 20644129.

32977340218432

Ans. 32068.

Extraction of the Nth Root of Numbers.

142. The nth root of a number is such a number as being taken n times as a factor will produce the given number, n being any positive whole number. When such a root can be exactly found, the given number is a perfect nth power; all other num

bers are

imperfect nth powers.

Let N denote any number whatever. If it is expressed by

less than

will be

n1 figures, and is a perfect nth power, its nth root

expressed by a single figure, and may be found by

means of a table containing the nth powers of the first ten numbers.

"

If the number is not a perfect nth power, it will fall between two nth powers in the table, and its root will fall between the nth roots of these powers.

If the given number is expressed by more than n figures, its root will consist of a certain number of tens and a certain number of units. If we designate the tens of the root by a, and the units by b, we shall have, by the binomial formula,

Now, as the nth power of the tens, cannot be less than 1 followed by n ciphers, the last n figures on the right, cannot make a part of it. They must then be pointed off, and the nth root of the greatest nth power in the number on the left will be the number of tens of the required root.

Subtract the nth power of the number of tens from the num ber on the left, and to the remainder bring down one figure of the next period on the right. If we consider the number thus found as a dividend, and take n times the (n - 1)th power of the number of tens, as a divisor, the quotient will evidently be the number of units, or a greater number.

If the part on the left should contain more than n figures, the n figures on the right of it, must be separated from the rest, and the root of the greatest nth power contained in the part on the left extracted, and so on. Hence the following

RULE.

I. Separate the number N into periods of n figures each, beginning at the right hand; extract the nth root of the greatest perfect nth power contained in the left hand period, it will be the first figure of the root.

II. Subtract this nth power from the left hand period and bring down to the right of the remainder the first figure of the next period, and call this the dividend.

III. Form the n 1 power of the first figure of the root, multiply it by n, and see how often the product is contained in the dividend: the quotient will be the second figure of the root, or something greater.

IV. Raise the number thus formed to the nth power, then subtract this result from the two left-hand periods, and to the new remainder bring down the first figure of the next period: then divide the number thus formed by n times the n- - 1 power of the two figures of the root already found, and continue this operation until all the periods are brought down.

We first point off, from the right hand, the period of four figures, and then find the greatest fourth root contained in 53, the first period to the left, which is 2. We next subtract the 4th power of 2, which is 16, from 53, and to the remainder 37 we bring down the first figure of the next period. We then divide 371 by 4 times the cube of 2, which gives 11 for a quotient but this we know is too large. By trying the numbers 9 and 8, we find them also too large. then trying 7, we find the exact root to be 27.

143. When the index of the root to be extracted is a multiple of two or more numbers, as 4, 6, . . . &c., the root can be ob tained by extracting roots of more simple degrees, successively. To explain this, we will remark that,

(a3)4 = a3 X a3 X a3 X a3

a3+3+3+3 = a3×4=

and, in general, from the definition of an exponent

(am)n = am × am × am × am = am xn:

a12,

hence, the nth power of the mth power of a number is equal to the mnth power of this number.

Let us see if the converse of this is also true.

then raising both members to the nth power, we have, from the definition of the nth root,

m√ a =
a = bn ;

and by raising both members of the last equation to the mth power

a = bmn

Extracting the mnth root of both members of the last equation,

since each is equal to b. Therefore, the nth root of the mth root of any number, is equal to the mnth root of that number. in a similar manner, it might be proved that

And

REMARK.-Although the successive roots may be extracted in any order whatever, it is better to extract the roots of the lowest degree first, for then the extraction of the roots of the higher degrees, which is a more complicated operation, is effected upon numbers containing fewer figures than the proposed number.

Extraction of Roots by Approximation.

144. When it is required to extract the nth root of a number which is not a perfect nth power, the method already explained, will give only the entire part of the root, or the root to within less than 1. As to the part which is to be added, in order to com plete the root, it cannot be obtained exactly, but we can approximate to it as near as we please.

Let it be required to extract the nth root of a whole number,

denoted by a, to within less than a fraction ; that is, so near,

1 p

If we denote by r the root of the greatest perfect nth power in

r

will be greater than the difference between and the true

p

is the required root to within less than the

To extract the nth root of a whole number to within less than

1

a fraction multiply the number by p2; extract the nth root of

p

the product to within less than 1, and divide the result by p.

Extraction of the nth Root of Fractions.

145. Since the nth power of a fraction is formed by raising both terms of the fraction to the nth power, we can evidently find the nth root of a fraction by extracting the nth root of both terms.