The following example will be sufficient to illustrate the course to be pursued, whatever the degree of the root required may be. Let it be proposed to extract the fifth root of the polynomial 32 a1o — 80 a3 b3 + 80 ao bε — 40 a1 b2 + 10 a2 b12 — b15 The proposed being arranged with reference to the powers of the letter a, we seek the fifth root of the first term 32 a1o. Its root 2 a2 will be the first term of the root sought. We write therefore 2 a2 in the place of the quotient in division, and subtracting its fifth power from the whole quantity, we have for a remainder -80 a8 b380 a6 b6+ &c. The second term of the binomial (a+b)5 is 5 a4b; this shows, that in order to obtain the second term of the root, we must divide. - 80 as b3, the second term of the proposed, by five times the 4th power of 2 a2, the term of the root already found. Performing the operation we obtain -63. This will be therefore the second term of the root. Raising 2 a2 — b3 to the fifth power, it produces the quantity proposed. The root is therefore obtained exactly. If the root contained more than two terms, it would be necessary to subtract the fifth power of 2 a2—b3 from the proposed quantity, and then in order to find the next term of the root, to divide the first term of the remainder by five times the 4th power of 2 a2 — b3. In this case, however, only the first term of the divisor would be used; we should have therefore the same divisor, that was used the first time. 141. When the index of the root has divisors the root may be found more readily than by the general method. Thus the fourth root may be found by extracting the square root twice successively; for the square root of a4 is a2, and that of a2 is a, the fourth root of a1. In general, all roots of a degree marked by 4, 8 or any power of 2 may be found by successive extractions of the square root. Roots, the indices of which are not prime numbers, may be reduced to others of a degree less elevated. The 6th root, for example, may be found by first extracting the square and then the third root; for the square root of as is a3, and the third root of a3 is a. EXAMPLES. 1. To find the third root of 8x+36 x2+54 x + 27. · 2. To find the third root of 26 +6 x 40 x 96 x2 — 64. - + 3. To find the third root of 15 x1-6x+x— 6 x3- 20 x3 + 15 x2 + 1. 4. To find the third root of 27 x 54 x 63 x1 — 44 x2 + 21 x2-6x+1. 5. To find the fourth root of 216 a2 x2 — 216 a x3 +81 x2 + 16 a1 — 96 a3 x. SECTION XVIII. CALCULUS OF RADICAL EXPRESSIONS. 142. Radical expressions, the roots of which cannot be found exactly, frequently occur in the solution of questions. On this account mathematicians have been led to investigate rules for performing upon quantities subjected to the radical sign, the operations designed to be performed upon their roots. In this way the calculations required in the solution of a question are frequently rendered more simple, and the extraction of the root is left to be performed at last, when the radical expression is reduced to the most simple form, which the nature of the question will allow. ADDITION AND SUBTRACTION. 143. Radical expressions of the same degree, and which have the quantities placed under the radical sign also the same, are said to be similar. The addition and subtraction of similar radicals is performed upon the coefficients. Thus the sum of the radicals 3 5 5 5 36,9 bis 12b; the sum of a, b b2 c, — c / b2 c 5 is (a+b-c) b2 c. 3 In like manner 9 ac subtracted from 12 ac gives 3 7 7 3√ aa c, and bNo a b2 subtracted from a a b2 gives 7 (a - b)√ a b2 Radical expressions, which are at first dissimilar frequently become similar when reduced to their most simple form. Thus, let it be required to add 5 √2 a b2 and a √54 a2 b2. These expressions reduced to their most simple form become 3 3 5 a √2 a2 b32, 3 a √2 a2 b2; their sum is therefore The addition and subtraction of dissimilar radicals can be effected only by means of the signs + and -. EXAMPLES. 1. To find the sum of 5 a2 √ b c2, and a 9 a b c2. 50. Ans. 82. 5. To find the sum of 8 √, — 1√12, 4√27, and — 2√ Ans. 223. 6. To find the sum of 2 √ √60, —√√/15, and ✔✅ 3. Ans. 28/15. 7. To find the sum of 18 a b3, and No50 a3 b3. Ans. (3 ab5ab)2 ab. 8. To subtract 9a2 b c from 7 ab2 c. 144. Let it be required to multiplya by b, we have axbab; for ab raised to the seventh 7 7 power gives a b for the result, and ab raised to the seventh power gives also ab for the result; whence the seventh pow ers of these expressions being equal, the expressions themselves must be equal. The same reasoning may be applied to all similar cases; we have therefore the following rule for the multiplication of radical expressions of the same degree, viz. Take the product of the quantities under the radical sign, observing to place the result under a sign of the same degree. 5 5 5 Let it next be required to divide ✔a by ✔b. 5 5 case we have ✔ a '; for the expressions 5 5 5 In this a being raised to the fifth power give each these expressions are therefore equal. We have then the following rule for the division of one radical quantity by another of the same degree, viz. Take the quotient arising from the division of the quantities under the radical sign, recollecting to place it under a sign of the same degree. 3 EXAMPLES. 3 1. Multiply 4,76, and 5 together. 3 Ans. 120. 2 together. Ans. 140. Ans. 317 6. 2. Multiply 53, 7, and 8. Divide 1 by 1+2. Expressing the quotient in the form of a fraction, and multiplying both terms by 1-2, we have 9. Divide 16 by 22/3. Ans. 2-1. Ans. 23. FORMATION OF POWERS AND EXTRACTION OF ROOTS. 5 145. Let it be required to raise the radical ab to the according to the rule established for multiplication. Whence to raise a radical quantity to any power; we raise the quantity placed under the radical sign to the power required, observing to place the result under the same radical sign. When the index of the radical is a multiple of the exponent of the power to which the radical is to be raised, it may be raised to the power required in a more simple manner than by the preceding rule. Thus let it be required to raise to the second power. The proposed from what has been said, art. 141, may be put under the form √√2a; but to raise this expression to the second power, it is sufficient to suppress the first radical sign; 4 whence (2a)2=√za. 9 Again, let it be required to raise 5b to the third power. 3 The proposed may be put under the form√56; whence (5b)=√56. 9 3 Whence if the index of the radical is divisible by the exponent of the power, to which the proposed quantity is to be raised, the operation is performed by dividing the index of the radical by the exponent of the power. With respect to the extraction of roots, it is evident from the preceding rules, that to extract the root of a radical, we may extract the root of the quantity placed under the radical sign, the result being left under the same radical sign, or we may multiply the index of the radical by the index of the root to be extracted. |