To find the Difference of Radical Quantities. 223. When the radicals are similar, it is evident that the subtraction may be performed in the same manner as addition, except that the signs in the subtrahend are to be changed. Hence we have the following RULE. Reduce each radical to its simplest form. If the resulting radicals are similar, find the difference of the coefficients, and to the result annex the common radical part. If they are dissimilar, the subtraction can only be indicated. To multiply Radical Quantities together. 224. We have found, Art. 215, that Va multiplied by V is equal to Vab. If the radicals have coefficients, the product of the coeffi cients may be taken separately. Thus, also, a√xxb√y=axb× √x × √y=ab√xy; If the radicals have not a common index, they must first be reduced to a common index. Hence we have the following RULE. If necessary, reduce the given radicals to a common index. Multiply the coefficients together for a new coefficient; also multiply the quantities under the radical signs together, and place this product under the common radical sign. Then reduce the result to its simplest form. 225. We have seen, Art. 58, that the exponent of any letter in a product is equal to the sum of the exponents of this letter in the multiplicand and multiplier. That is, aTM×a"=aTM+", where m and n are supposed to be positive whole numbers. When one or both of the exponents are negative, we must take the algebraic sum of the exponents. For, suppose n is negative. Then The same relation holds true when m and n are fractional; that is, P For aïxa2=Va1×Vỡ, Art. 214,=VaTM×Va, Art. 220, Hence we conclude that the exponent of any letter in a product is equal to the algebraic sum of the exponents of this letter in the multiplicand and multiplier, whether the exponents are positive or negative, integral or fractional. 4. Find the product of a3, a3, a‡, and a ̄13. Ans. 15a Ans. 63a 63a1⁄4 Multiplication of Polynomial Radicals. 226. By combining the preceding rules with that for the multiplication of polynomials, Art. 61, we may multiply together radical expressions consisting of any number of terms. Ex. 1. Let it be required to multiply aa+2a_a* by a*—3a+2. Ans. 41. Ex. 4. Multiply 9+2√10 by 9-2√10. Ex. 5. Multiply 3√45–7√5 by √1+2√9%. Ans. 34. Ex. 6. Multiply c√a+d√b by c√a—d√b. Ans. ac2-bd2. Ex. 7. Multiply a3—a3+aa—a2+aa—a+a3—1 by a3+1. Ans. a1-1. To divide one Radical Quantity by another. 227. The division of radical quantities depends upon the following principle: The quotient of the nth roots of two quantities is equal to the nth root of their quotient; or, Va Γα for the nth power of each of these expressions is, Art. 186. Let it be required to divide 4a2√bby by 2a√3b. = 4a2√bby_4a2/6by Hence we have the following RULE. =2a√2y, Ans. If necessary, reduce the given radicals to a common index. Divide the coefficient of the dividend by that of the divisor for a new coefficient; also the quantity under the radical sign in the dividend by that in the divisor, and place this quotient under the common radical sign. Then reduce the result to its simplest form. 228. We have seen, Art. 72, that the exponent of any letter in a quotient is equal to the difference between the exponents of this letter in the divisor and dividend. The same relation holds true whether the exponents are positive or negative, integral or fractional; that is, universally, For the quotient must be a quantity which, multiplied by the divisor, shall produce the dividend; and, according to Art. 225, the exponent of any letter in a product is in all cases equal to the algebraic sum of the exponents of this letter in the multiplicand and multiplier. Hence this relation must hold true universally in division. Division of Polynomial Radicals. 229. By combining the preceding rules with that for the division of polynomials, Art. 80, we may divide one radical expression by another containing any number of terms. |