SECTION VI.-Evolution. 1. To find the roots of monomials.-Extract the corresponding root of the coefficient for the new coefficient: then multiply the index of the letter or letters by the index of the root, the result will be the exponents of the letter or letters to be placed after the coefficient for the root required. Examples. 1. Find the fourth root of 81 aa z3. First 819=3, new coefficient. Then 4 × = 1, exponent of a, and 8 × = 2, exponent of z. Hence 3 a z3 is the root required. 2. To find the square root of a polynomial.-Proceed as in the extraction of the square root, in arithmetic. Examples. 1. Extract the square root of a + 4 a3 x + 6 a® x2 + 4 ax3 +2°. a + 4 a3 x + 6 a2x2 + 4 a x3 + x^ (a2 + 2 a x + x2 2 a2+2 ax) 4 a3 x + 6 a2x2 4 a3 x + 4 a3 xo 2 a3 + 4 a x + x2) 2 a2 x2 + 4 a x3 +24 2 a3 x2 + 4 a x3 x x1 2. Extract the square root of x1 — 2 x3 + 2x2 — } x + 15 1 (x2 2 16 x + root. X4 3. To find the roots of powers in general.—If they be not the roots of high powers that are required, the following rule may be employed: Find the root of the first term, and place it in the quotient. -Subtract its power, and bring down the second term for a dividend.-Involve the root, last found, to the next lowest power, and multiply it by the index of the given power for a divisor.-Divide the dividend by the divisor, and the quotient will be the next term of the root.-Involve the whole root, and subtract and divide as before; and so on till the whole is finished. [+ 1 root req. 26-6x+15-20 x3+15 x2-6x+1)x2-2 x 2. Find the 4th root of 16 a1 ·96 a3 x + 216 a3 x2. Note. In the higher roots proceed thus: For the biquadrate, extract the square root of the square root. sixth root ninth root, cube root of the square root. sq. rt. of the sq. rt. of the sq. rt the cube root of the cube root. Examples, however, of such high roots seldom occur in any practical inquiries SECTION VII.-Surds. A Surd, or irrational quantity, is a quantity under a radical sign or fractional index, the root of which cannot be exactly obtained. (See ARITH. Sect. 9. Evolution.) Surds, as well as other quantities, may be considered as either simple or compound, the first being monomials, as √3, a3, a b3, the others polynomials, as √3 + √5,ŵa + √b -✓cs, &c. Rational quantities may be expressed in the form of surds, and the operation, when effected, often diminishes subsequent labour. Reduction. 1. To reduce surds into their simplest expressions. 1. If the surd be not fractional, but consist of integers or integral factors under the radical sign: Divide the given power by the greatest power, denoted by the index, contained therein, that measures it without remainder; let the quotient be affected by the radical sign, and have the root of the divisor prefixed as a coefficient, or connected by the sign X. Examples. 1. √75= √(25 × 3) = √25 × √3 = 5 √5 2. 3. 176 = √(8x 12 x2 y) = √4 x3 (2 x — 3 y) = √4x3 × 3 y) 4. ✓ (2 x 5. 108 x3y = 6. = 3xy 4y. (27 x3 y3 × 4 y)=27 x3 y3 × 43' (56 x3 y + 8 x3) = (7y+ 1) = 2 x 8 x3 (7 y + 1) = 8 x3 x (7 y + 1). 2. If the surd be fractional, it may be reduced to an equivalent integral one, thus : Multiply the numerator of the fraction under the radical sign, by that power of its denominator whose exponent is. 1 less than the exponent of the surd. Take the denominator from under the radical sign, and divide the coefficient (whether unity, number, or letter) by it, for a new coefficient to stand before the surd so reduced. Note. This reduction saves the labour of actually dividing by an approximated root; and will often enable the student to value any surd expressions by means of a table of roots of integers. Examples. 号 1. √} = √(} • }) = √3 = √÷ × √3 = }√3. 2. √} = √(} · §) = √1⁄2 = √13 × √5 =} √ 5. 3. If the denominator of the fraction be a binomial or residual, of which one or both terms are irrational and roots of squares : Then, multiply this fraction by another which shall have its numerator and denominator alike, and each to contain the same two quantities as the denominator of the given expression, but connected with a different sign. Note. By means of this rule, since any fraction whose numerator and denominator are the same, is equal to unity, the quantity to be reduced assumes a new appearance without changing its value; while the expression becomes freed from the surds in the denominator, because the product of the sum and difference of two quantities is equal to the difference of their squares. n-1 (5a — 5o 34 + 5† 3a — 3o) Note 2.-Upon the same general principle any binomial or residual surd, as A+B may be rendered rational by taking A (AB) + (A"-3 B) ~ (A" B3) + &c. for a multiplier: where the upper signs must be taken with the upper, the lower with the lower, and the series cor tinued to n terms. Thus, the expression ab3, multiplied by ✔a + ✅a° b3 + a b + b, gives the rational product a3-b2 |