3. Divide the dividend by the divisor, and annex the result both to the quotient and to the divisor. 4. Multiply the divisor, thus increased, by the term last set in the quotient, and subtract the product from the dividend. And so on, always the same, as in common arithmetic. 2. Find the root of a + 4a3b + 10a2b2 + 12ab3 +961. a' + 4a3b + 10a2b2 + 12ab3 + 9b1 (a2 + 2ab + 3b3. a*. 2a+2ab) 4a3b + 10a2b2 4a3b+4a2b2 2a2 + 4ab + 3b2) 6a2b2 + 12ab3 + 9b1 6a2b2+12ab3 + 9b1 3. To find the square root of a1 + 4a3 + 6a2 + 4a + I. Ans. a2a + 1. 4. Extract the square root of a1 2a32a2 -a a+1. Ans. aa + 1. 5. It is required to find the square root of a2 ab. b3 b b2 Ans. a CASE III. To find the Roots of any Powers in general. THIS is also done like the same roots in numbers, thus: Find the root of the first term, and set it in the quotient. --Subtract its power from that term, and bring down the second term for a dividend.--Involve the root, last found, to the next lower power, and multiply it by the index of the given power, for a divisor.-Divide the dividend by the divisor, and set the quotient as the next term of the root.Involve now the whole root to the power to be extracted; then subtract the power thus arising from the given power, and divide the first term of the remainder by the divisor first found; and so on till the whole is finished *. EXAMPLES.. 1. To find the square root of a1—2ab+3a2b3—2ab3+b'. a1—2ab+3a2b2 — Qab3 + b1 (a2 — ab + b2. a'—2ab+3a2b2 — Qab3 + b2 = (a2 — ab + b2)2. 2. Find the cube root of ao - 6a3 + 21a1-44a3 + 63a2 54a + 27. - 6a3 +21a1 — 44a3 + 63a2· · 54a +27 (a2-2a+3. a-6a+21a-44a+63a2-54a+27= (a2-2a+3). * As this method, in high powers, may be thought too laborious, it will not be improper to observe, that the roots of compound quantities may sometimes be easily discovered, thus: Extract the roots of some of the most simple terms, and connect them together by the sign+or, as may be judged most suitable for the purpose.-Involve the compound root, thus found, to the proper power; then, if this be the same with the given quantity, it is the root required. But if it be found to differ only in some of the signs, change them from to, or from to, till its power agrees with the given one throughout. Thus, in the 5th example, the root 3a 2b, is the difference of the roots of the first and last terms; and in the 3d example, the root eb, is the sum of the roots of the 1st, 4th, and 6th terms. The same may also be observed of the 6th example, where the root is found from the first and last terms. 3. To find the square root of a3-2ab+ 2ax + b2. 2bx+x2. .4. Find the cube root of a 12a + 8. Ans. ab + x. 3a59a1-13a2 + 18a2 Ans. a2-a + 2. 5. Find the 4th root of 81a1-216a3b +216a2b2 +1661. 6. Find the 5th root of a5 10a40a3 - 32. 7. Required the square root of 1 - x2. 8. Required the cube root of 1 — x3. Ans. 3a-26. 80a2 + 80a Ans. a -2. SURDS. SURDS are such quantities as have no exact root; and are usually expressed by fractional indices, or by means of the radical sign✓. Thus, or 3, denotes the square root of 3; and 2, or 3/23, or 3/4, the cube root of the square of 2; where the numerator shows the power to which the quantity is to be raised, and the denominator its root. PROBLEM I. To reduce a Rational Quantity to the Form of a Surd. RAISE the given quantity to the power denoted by the index of the surd; then over or above the new quantity set the radical sign, and it will be of the form required. EXAMPLES. 1. To reduce 4 to the form of the square root. 2. To reduce 3a to the form of the cube root. then 27a or (27a") is the answer. 3. Reduce 6 to the form of the cube root. Ans. (216) or /216. 4. Reduce ab to the form of the square root. Ans. ✔ab. 5. Reduce 2 to the form of the 4th root. 6. Reduce to the form of the 5th root. 7. Reduce a +x to the form of the square root. 8. Reduce ax to the form of the cube root. Ans. (16)+. PROBLEM II. To reduce Quantities to a Common Index. 1. REDUCE the indices of the given quantities to a common denominator, and involve each of them to the power denoted by its numerator; then 1 set over the common denominator will form the common index. Or, 2. If the common index be given, divide the indices of the quantities by the given index, and the quotients will be the new indices for those quantities. Then over the said quantities, with their new indices, set the given index, and they will make the equivalent quantities sought. Therefore 31% and 5% = (35) and (52)T'% = 105 and 10/5" ='/243 and 1/25. 2. Reduce a3 and b3 to the same common index §. Here, the 1st index, = and ÷ = × = the 2d index. 1 Therefore (a) and (b3)3, or √✅a' and √/b} ties. are the quanti 3. Reduce 4 and 5 to the common index ¿. Ans. (256) and 25*. 4. Reduce a3 and x to the common index ¿. Ans. (a2)a and (x3)3. 5. Reduce a2 and r3 to the same radical sign. PROBLEM III. To reduce Surds to more Simple Terms. DIVIDE the surd, if possible, into two factors, one of which is a power of the kind that accords with the root sought; as a complete square, if it be a square root, a complete cube, if it be a cube root; and so on. Set the root of this complete power before the surd expression which indicates the root of the other factor; and the quantity is reduced, as required. If the surd be a fraction, the reduction is effected by multiplying both its numerator and denominator by some number that will transform the denominator into a complete square, cube, &c. its root will be the denominator to a fraction that will stand before the remaining part, or surd. See Example 3, below. EXAMPLES. 1. To reduce 32 to simpler terms. Here 32 = (16×2) = √16 × √2=4 × √2 = 4√√2. 2. To reduce /320 to simpler terms. 3/320/(64× 5) 64 X 54 × 2/5 4/5. 3. Reduce to simpler terms. Note. There are other cases of reducing algebraic surds to simpler forms, that are practised on several occasions ; one of which, on account of its simplicity and usefulness, may be here noticed, viz. in fractional forms having compound surds in the denominator, multiply both numerator and denominator by the same terms of the denominator, but having one sign changed, from to or from to +, which will reduce the fraction to a rational denominator. |