If the given surds are of different orders, they must first be transformed into equivalent surds of the same order. Multiply the coefficients together for a new coefficient, and the expressions under the radical signs for a new radicand, and reduce the result to simplest form. Ex. 1. 5√√6 × 3√√2 = 5·3√/6⋅ 2 = 15√/T2 = 30√√/3. Ex. 2. 5√/5 × √2 = √/5a × √/22 = √53 × 22 = √500. It is often convenient to obtain the prime factors of the radicands before multiplying. Write each of the following products in simplest form : MULTIPLICATION OF POLYNOMIALS INVOLVING SURDS 37. The product of two polynomials the terms of which contain surds may be obtained as follows: Multiply each term of the multiplicand by each term of the multiplier. Ex. 1. (√√/2−5√/3+√/6−7)×3√/6=3√/12–15√/18+31/36–211/6 6/345/2+18 - 21√/6 =18-45√√√2 +6√/3 – 21√√√6. Ex. 2. Multiply 5√5 + 2√2 by 4√/5 − 3√√2. 38. Two binomial quadratic surds which differ only in the sign. of one of the surd terms are called conjugate surds. E. g. Va+√b and Va√b; 5+4/3 and 5-41/3: √6+ √/7 and √6+ √7. 39. The product of two conjugate surds is a rational number or expression. Representing two conjugate surds by √x+ √y and √x- √y, we have, (√x+ √ÿ){√x − √ÿ) = {√x)2 − (√/y)2 = x − y. Ex. 3. (2√/7 + √/II)(2√/7 – √/TT) = 28 – 11 = 17. INVOLUTION OF MONOMIAL SURDS 40. From the principle (Va)" = √a", it follows that: An entire monomial surd may be raised to a power by raising the radicand to the indicated power. 41. The quotient obtained by dividing one entire monomial surd by another may be found by applying the principle The process for finding the quotient of one mixed surd divided by another may be made to depend upon the principle above. Since two surds of different orders can be transformed into equivalent surds of the same order, it follows that it is necessary to state the process only for mixed surds of the same order. The index of the indicated root of the radicand of the quotient obtained by dividing one monomial surd by another of the same order is equal to the common index of the indicated roots of the radicands of the dividend and divisor. For the coefficient of the radical part of the quotient divide the coefficient of the dividend by the coefficient of the divisor, and for the radicand of the quotient divide the radicand of the dividend by the radicand of the divisor. The result should be reduced to simplest form. |