being contained in 6a two times, it follows that 2 must be the third term of the root. The first correction of the second incomplete divisor is (2) × 3 × 2 = 6a2 — 6x, and (2) = 4 is the second correction; consequently, 3 — 6×3 + 9x2 -6x+4 is the second complete divisor. Because the product of the second complete divisor and 2, when subtracted from the second remainder, gives 0 for the remainder, it follows that the root is exact. 2. To extract the cube root of a3 + 3a2b+3ab2 + b3 +3a2c +6abc + 3b°c + 3ac2 + 3bc2 + c3. Ans. a + b + c. 3. To extract the cube root of 27 + 54x2 + 36x1 + 8xo. 7. To find the cube root of 8x + 36x + 138x + 279x2+ 483x2 + 441x + 343. Ans. 2x2 + 3x + 7. Final Remarks.-The beautiful method of Horner, as exhibited in the scheme (E), given at p. 115, can be applied to the extraction of roots of all degrees, and that, whether the expressions whose roots are to be extracted are algebraic or arithmetical; observing, that an algebraic expression must be arranged according to the powers of one of its letters. EXAMPLES. Ex. 1.-To extract the square root of 4x-4x1 + 12x3 + x2 6x + 9. Let y stand for the root, then we shall have y2 = 4x6 — a6x9= 0. Hence, detaching the coefficients of the powers of y, since the root of 4 is 2x3, we shall by the method get 1+0 -4x+0+4x-12x3-x2+6x-9|2x3 +0−x+3=2x3 Ex. 2.-To extract the cube root of a 6a+ 15α1 20a3+15a6a+ 1. Representing the root by y, we easily get the equation y3 +0. y2+0.y — a + 6a — 15a+20a3 - 15a2 + 6a - 1 0. Since a is the root of the first term, by detaching the coefficients, etc., we shall get 1+0+ 0 -a®+6a3-15a1+20a3—15a2 + 6a−1|a2—2a+ a1+a®-6a+12a1- 8a3 1 = the a2 a2 root. 2a -3a1+12a3-15a2+6a—1 3a2-6a+1 3a1-12a3+15a2—6a+1 It is clear that we may proceed in a similar way to what we have done in these examples, in order to extract any algebraic root; and it is also easily seen how we may, by the same method, raise any algebraic expression to any proposed power. We propose to show how to apply the same method to the extraction of arithmetical roots, in the remaining examples. Ex. 3.-To extract the square root of 56169. Because 56169 = 10000 x = 56169 = 10000 (100) 5.6169, we shall have 56169 100 5.6169; consequently, if we extract the root of 5.6169, and move the decimal point in the root two places to the right, we shall get the root of 56169. 5.61.69, that we shall have 56169 100 /5.61.69, which is in accordance with the rule in Arithmetic for pointing off the number, preparatory to the extraction of its root. Proceeding as in the preceding questions, to extract the root of 5.6169, we shall get The analogy of this method of extracting the root to the common methods is so evident as scarcely to need any com ment. For 2 is the root of the greatest square in 5, 4 is the first incomplete divisor, and 1.61 the corresponding dividend; 4.3 is the first complete divisor, which, multiplied by .3, the second figure of the root, gives the product 1.29; and 4.6 is the second incomplete divisor, and 0.3269 the corresponding dividend, etc. Ex. 3. To extract the cube root of 41781923. 41781923 = 100 41.781923 10041.781.923. To extract the cube root of 41.781.923, we have = -41.781.923 | 3.47 the root; and 347 is the ond dividend, its incomplete and complete divisors being complete and complete divisors; also, 2.477.923 is the secHere the first dividend is 14.781, and 27, 30.76 are its in 34.68 and 35.3989. places. Proceeding as in the preceding examples, we get In this example, 6 is the first dividend, and 5, 10.9456 are its incomplete and complete divisors; 1.62176 is the second dividend, and 19.2080, 21.22726501 are its incomplete and complete divisors, and so on. Having found the first four decimals as in the preceding examples, we have found the remaining three by dividing the dividend + 0.00173495 + by its incomplete divisor 23.7116794 —. SECTION XII. SURDS, OR IRRATIONAL EXPRESSIONS. (1.) WHEN a root is indicated by the use of the surd sign or by a fractional index, and is such that the root can not be exactly extracted, it is called a surd, or an irrational expression. Thus, 12, 14, 8, Va3, vb, 38, and Vabc are surds, because the indicated roots can not be exactly extracted. (2.) Any number or quantity which is not affected by the sign of any root, or which can be freed from the sign of any root by the exact extraction of the root, is said to be rational. Thus, 3, 5, 7, a, bc, def, are rational; and so are 36, V27, Vas, 10, since their roots are ± 6, 3, ± a2, and b2, rejecting imaginary roots. (3.) If a monomial consists of two factors, such that one of them is rational, while the other is irrational, then the rational factor, when written before or to the left of the irrational factor, is called its coefficient. Thus, in 5 √2 we call 5 the coefficient of √2, and in abcd, ab is called the coefficient of Ved. If a monomial surd has no coefficient expressed, it is clear that the coefficient must be understood to be 1. Thus, 5 is clearly the same as 1 x 5, and ab = 1 x Wab. m (4.) Monomial surds, which do not differ from each other in any other respect than their coefficients and signs, are |