inferior local value. The remaining part of the operation requires no further explanation. It will be seen that the number of places in the root is equal to the number of places assumed in the power. Hence we have the following RULE.-I. Assume as many places in the power as there are places required in the root, and proceed in the usual manner till all the assumed figures have been brought down. II. Form the next trial divisor as usual, omitting the ciphers at the right; and reject one place in forming each subsequent trial divisor. III. In completing the first contracted divisor, omit to annex the new figure of the root to the corresponding term in the left-hand column, and reject two places in forming each succeeding term in this column. IV. In multiplying, treat the right-hand figure of each contracted term as redundant, both in the column at the left, and in the column of divisors. NOTE.-To avoid complicating the process of contracting, it is better to use none but full periods, even if the root is thereby carried beyond the required number of places. 1. Find the cube root of 3 correct to 6 decimal places. Ans. 1.442249+. 2. Find the cube root of 7 correct to 6 decimal places. Ans. 1.912931+. 3. Find the cube root of 156 correct to 8 decimal places. Ans. 5.38321261+. 4. Find the cube root of 34786 correct to 6 decimal places. Ans. 32.6438594. 5. Find the cube root of 10.973937 correct to 6 decimal places. Ans 2.222222 4. 6. Find the cube root of 1500.101520125 correct to 8 decimal places. Ans. 11.44740066+. 7. Find the cube root of 1.164132 correct to 6 decimal places. Ans. 1.051963+. SECTION IV. RADICAL QUANTITIES. 244. A Radical Quantity is a root merely indicated, either by the radical sign or by a fractional exponent; as 3√a, Va−b, c(a+b)3, m√x2 — y2. A radical quantity may be either surd or rational. The quantity or factor placed before a radical is its coefficient. Thus in the examples just given, 3, 1, c, and m are the coefficients of the radicals. 245. The Degree of a radical quantity is denoted by the radical index, or by the denominator of the fractional exponent. Thus, √a, (a - b) V x2-y, n are radicals of the 2d degree; are radicals of the 3d degree; Vac, (x+y) are radicals of the nth degree; 246. Similar Radicals are those in which the same quantity is affected by radical signs having the same index. Thus, 4√ a2 + b, Va2+b, and 7(a2 + b) are similar radicals. REDUCTION OF RADICALS. CASE I. 247. To reduce a radical to its simplest form. A radical is in its simplest form when it contains no perfect power corresponding to the degree of the radical. 1. Reduce 48ax3 to its simplest form. We have seen that the nth root of a quantity is equal to the product of the nth roots of its component factors (227). Hence we have √48a6x3 = √16ax x 3x= √16a6x2 × √/3x=4a3x √3x. It will be seen that we first separate the quantity under the radical sign into two factors, one of which is a perfect square. Then according to the principle of evolution just adduced, we have the product of two radicals, one of which, 16ax2, is rational, and the other, √3x, is a surd. The expression is then reduced by extracting the root of the rational part, and multiply. ing it by the surd. · 2. Reduce 38x+ys — 8x3y1 to its simplest form. Factoring as before, we have 3√/ 8x1y3 — 8x3y1 = 3 × √ 8x3y3 × х y y RULE.-I. Separate the factors of the quantity under the radical sign into two groups, one of which shall contain all the perfect powers corresponding in degree to the radical. II. Extract the root of the rational part, multiply this root by the given coefficient, and prefix the product to the surd or irrational part. 248. When the quantity under the radical sign is a fraction, we may transform it in such a manner that the denominator shall be a perfect power corresponding in degree to the indicated root. Then after simplifying, the quantity remaining under the radical sign will be entire. It will generally be expedient to separate the given fraction into two factors, one of which shall be a perfect power; we may then operate upon the surd part separately. 1. Reduce to its simplest form. √ 11 OPERATION. = √ x = √ √ × 33 = √ √××33 = √33, Ans. 2. Reduce to its simplest form. 3 OPERATION. 3 V&2 = Vπ = Vætt × 15 = ††15, Ans.. In like manner reduce the following: CASE II. 249. To reduce a rational quantity to a radical, or to introduce a coefficient of a radical under the radical sign. Since involution and evolution are the converse of each other, we have a = √ a2 = √ a3 = √ a1, etc. Whence, we have also, a√b = √a2 × √b = √ a2b. We have, therefore, the following RULE.-I. To reduce a rational quantity to a radical :-Involve it to a power denoted by the degree of the required radical, and write the result under the radical sign. II. To introduce the coefficient of a radical quantity under the radical sign :-Involve it to a power denoted by the degree of the radical, and multiply the quantity under the radical by the power thus obtained. 1. Reduce ab2 to a radical of the second degree. Ans. √a2b1. 2. Reduce 5a2xy3 to a radical of the 3d degree. Ans. 125axy9. 3. Reduce a cz to a radical of the 4th degree. ́Ans. (a1 — 4a3cz + 6a2c2z2 — 4αc31⁄23 + c^zs). Introduce the coefficients of the following radicals under the radical sign: 4. 4a√2xy. 6. (a-2b)√2a. |