3. (a+b)vab= v(a+b)3× ab = √a3b+2a2b2+ab3. To reduce radicals of different indices, to equivalent radicals, having a common index. RULE. (57.) Find the least common multiple of all the indices; this will be the common index. Divide this common index by each particular index, and raise the quantity under the radical sign, to the power denoted by the quotient. 2— Let it be required to reduce Vac, v3b, and vab, to equivalent radicals having a common index. The least common multiple of the indices is 12; and if we divide this by each particular index, we obtain, for the quotients, respectively, 6, 4, and 3. The first radical is therefore to be raised to the sixth power, the second to the fourth, and the third to the third, and they become, Vac, v81b1, and Vab3. This rule depends upon the principle, that the value of a radical is not altered, when the index of the radical, and the exponent of the quantity under the sign, are both multiplied by the same number. To prove this, we have only to show mn m that the Va" is the same as the Va. We have seen (45) that the root of a quantity may be extracted by dividing the exponent of the quantity by the index of the root. Accord ing to this notation, the mnth root a", is written amn, which 3 15 Ans. Vabc and 6 8c6 a3 4. Reduce va, vb, c, and Vd to a common index. 3 Ans. Va, Vb1, Vc3, and √d2. 5. Reduce, and 1/2 to a common index. Ans. 1296, 6561, and V8. 6. Reduce 3 and 5 to a common index. 12 12 Ans. 150 and 151. ADDITION AND SUBTRACTION OF RADICALS. RULE. (58.) If the radicals are similar, add or subtract their coefficients; but if they are not similar, the addition or subtraction can only be indicated. EXAMPLES. 1. Find the sum of a √2ac, 2√2ac, and 3a √2ac. Ans. (4a-2) v2ac. 2. Find the sum of 3a √2bc, 7 V11a2, and 12b √19c2. Ans. 3a v2bc +7 VIIa2 + 12b v19c2. 3 3 3. Find the difference between 2b Vab and a va2b. Ans. (2b — a) va b. 4. Add together 3 Vab3c2, c Vab3, and 4b Vabc2. These radicals, in their present form, are not similar; but by reducing them to their simplest form, they become, respectively, 3bc Vab, bc Vab, and 4bc √ab. Their sum is, therefore, 8bc Vab. 5. From 11 96a2bc3 subtract 3a 1/24bc3. Ans. 38ac V6bc. 6. Find the sum of 4va +x and v4a3b2+4a2b2x. Ans. (4+2ab)va+x. (59.) Reduce the radicals to a common index, and then multiply the quantities under the sign, together, and place the product under the common radical sign. If there are coefficients, these must also be multiplied together. This rule depends upon the principle demonstrated in art. (56), that the nth root of the product of two or more quantities, is equal to the product of the nth roots: thus, for, raising both quantities to the nth power, we have, ax bab. EXAMPLES. 1. Multiply 3av2bc by 5b√3c. 3av2bc5bv3c = 15ab√2bc ×3c = 15ab√6bc2. 2. Multiply √3ab by 2v7a2c. Reduced to a common index, these radicals become 6 6 √27a3b3, and 249a*c*; and v (60.) Reduce the radicals to a common index; then divide the quantities under the sign, and place the quotient under the common radical sign. If the radicals have coefficients, the coefficient of the dividend must be divided by that of the divisor. These radicals, reduced to a common index, are √64, and (61.) Involve both the coefficients of the radical, and the radical itself, to the required power. Thus, (3va)3 = 9va3, and by reducing, = 9ava; 3 and, (10√3a2b)2 = 100√9a1b2 — 100a9ab2. This rule results directly from the rule for radicals; for va X va Xva — (va)3 iplication of |