the number of decimal places is equal to three times the number required in the root. The question is then reduced to extracting the cube root of a decimal fraction.

219. Suppose it is required to find the sixth root of 23, to within 0.01.

Applying the rule of Art. 215 to this example, we multiply 23 by 1006, or annex twelve ciphers to 23; then extract the sixth root of the number thus formed to within unity, and divide this root by 100, or point off two decimals on the right.

Extraction of Roots of Algebraic Quantities.

220. Before extracting the root of an algebraic quantity, let us see in what manner any power of it may be formed.

Let it be required to form the fifth power of 2a3b2. We have (2a3b2)5 = 2a3b2 × 2a3b2 × 2a3b2 × 2a3b2 × 2ạ3b2,

from which it follows, 1st. That the co-efficient 2 must be multiplied by itself four times, or raised to the 5th power. 2d. That each of the exponents of the letters must be added to itself four times, or multiplied by 5.

Hence,

In like manner,

(2a362)5 = 25. a3×562×5 = 32a15b10.

(8a2b3c)3 = 83. a2×363×3c3 = 512a6b9c3.

Therefore, in order to raise a monomial to any power, raise the co-efficient to this power, and multiply the exponent of each of the letters by the exponent of the power, and unite the terms. Hence, to extract any root of a monomial,

1st. Extract the root of the co-efficient and divide the exponent of each letter by the index of the root. 2d. To the root of the co

efficient annex each letter with its new exponent, and the result will be the required root. Thus,

3 64a9b3c64a3bc2; 16a8b12c42a2b3c.

From this rule we perceive, that in order that a monomial may be a perfect power, 1st, its co-efficient must be a perfect power; and 2d, the exponent of each letter must be divisible by the index of the root to be extracted. It will be shown hereafter, how the expression for the root of a quantity, which is not a perfect power, is reduced to its simplest terms.

221. Hitherto, in finding the power of a monomial, we have paid no attention to the sign with which the monomial may be affected. It has already been shown, that whatever be the sign of a monomial, its square is always positive.

Let n be any whole number; then, every power of an even degree, as 2n, can be considered as the nth power of the square; that is, (a2)" a2n.

=

Hence, it follows, that every power of an even degree, will be essentially positive, whether the quantity itself be positive or negative. Thus,

(±2a2b3c) = +16a8b12c4.

Again, as every power of an uneven degree, 2n + 1, is but the product of the power of an even degree, 2n, by the first power; it follows that, every power of an uneven degree, of a monomial, is affected with the same sign as the monomial itself.

Hence, (+4a2b)3 = +64a6b3; and (— 4a2b)3 — — 64a6b3

:

From the preceding reasonings, we conclude,

=

1st. That when the degree of the root of a monomial is uneven, the root will be affected with the same sign as the monomial.

2d. When the degree of the root is even, and the monomial a positive quantity, the root is affected either with the sign + or

Thus,

/81a4b123ab3;

6

64a18±2a3.

3d. When the degree of the root is even, and the monomial negative, the root is impossible; for, there is no quantity which, being

1

raised to a power of an even degree, will give a negative result. Therefore,

are symbols of operation which it is impossible to execute. They are imaginary expressions (Art. 126), like

Extraction of Roots of Polynomials.

222. Let us first examine the law of formation of any power of a polynomial. To begin with a simple example, let us develop (a + y + z)3.

If we place y + zu, we shall have,

(a + u)3 = a3 + 3a2u + 3au2 + u3 ;

or by replacing u by its value, y + z,

(a + y + z)3 = a3 + 3a2 (y + z) + 3a (y + z)2 + (y + z)3 ;

or performing the operations indicated,

' (a + y + z)3 = a3 + 3a2y + 3a2z + 3ay2+ 6ayz + 3az2 + y3+ 3y2z

+ 3yz2 + 23.

When the polynomial is composed of more than three terms, as

(a + u)3 = a3 + 3a2u + 3au2 + u3 ;

from which we see, that the cube of any polynomial is equal to the cube of the first term, plus three times the square of the first term multiplied by each of the remaining terms, plus other terms.

If u does not contain a, it is plain that the exponent of a in each term, as a3, 3a2u, &c., will be greater than in any of the following terms; and hence, every term will be irreducible with the terms which precede or follow it.

If u contains a, as in the polynomial

a2 + ax + b, where u = ax + b,

the terms will still be irreducible with each other, provided we arrange the polynomial with reference to the letter a. For, if the given polynomial be arranged with reference to a, the exponent of a in the first term will be greater than the exponent of a in u: hence, its cube will contain a with a greater exponent than will result from multiplying its square by u. Also, the co-efficient of u multiplied by the first term of u, will contain a to a higher power than any of the following terms of the development, and hence, will be irreducible with them; and the same may be shown for the subsequent terms.

In order to extract any root of a polynomial, we will first explain the method of extracting the cube root. It will then be easy to generalize this method, and apply it to the case of any root whatever.

Let N be any polynomial, and R its cube root. Suppose the two polynomials to be arranged with reference to some letter, as a, for example. It results from the law of formation of the cube of a polynomial (Art. 222), that in the cube of R, the cube of the first term, and three times the square of the first term by the second, cannot be reduced with each other, nor with any of the following terms.

Hence, the cube root of that term of N which contains a, affected with the highest exponent, will be the first term of R; and the second term of R will be found by dividing the second term of N by three times the square of the first term of R.

By examining the development of the trinomial a + y + z, we see, that if we form the cube of the two terms of the root found as above, and subtract it from N, and then divide the first term of the remainder by 3 times the square of the first term of R, the quotient will be the third term of the root. Therefore, having arranged the terms of N, with reference to any letter, we have, for the extraction of the cube root, the following

RULE.

I. Extract the cube root of the first term.

II. Divide the second term of N by three times the square of the first term of R; the quotient will be the second term of R.

III. Having found the first two terms of R, form the cube of this binomial and subtract it from N; after which, divide the first term of the remainder by three times the square of the first term of R: the quotient will be the third term of R.

IV. Cube the three terms of the root found, and subtract the cube from N: then divide the first term of the remainder by the divisor already used, and the quotient will be the fourth term of the root: the remaining terms, if there are any, may be found in a similar

manner.

EXAMPLES.

1. Extract the cube root of x6 — 6x5+15x1- 20x3+ 15x2-6x+1. x6 — 6x5+ 15x1 — 20x3 + 15x2 — 6x+1 | x2

(x2 — 2x)3 = x6 — 6x5 + 12x1 — 8x3

3x4

2x + 1

1st rem.

3x-12x3 +, &c.

(x2-2x+1)3x6 6x5+15x1-20x3+15x2-6x+1.

6x5,

2x times.

In this example, we first extract the cube root of x, which gives x2, for the first term of the root. Squaring x2, and multiplying by 3, we obtain the divisor 3x4: this is contained in the second term Then cubing the root, and subtracting, we find that the first term of the remainder 3x+, contains the divisor once. Cubing the whole root, we find the cube equal to the given polynomial. Hence, 2-2x+1, is the exact cube

root.

223. The rule for the extraction of the cube root is easily extended to a root with a higher index. For,

Let

a+b+c+...f, be any polynomial.

Let s the sum of all the terms after the first.

Then a + s = the given polynomial; and

1

(a + s)n = =an + nan-1s+ other terms.

That is, the nth power of a polynomial, is equal to the nth power of the first term, plus n times the first term raised to the power