2. The product 3a1 / 8a2 × 2b 1 / 4a2c = 6ab + / 32a*c = 12a2b 4 / 2c. 3. The product 4. The product 5. 3 Multiply √√3 by √ √ 6. Multiply 2/15 by 33/10. 5 7. Multiply 4 by 2 Ans. 12/8. Ans. 66/337500. 8. Multiply√2,3/3, and 5, together. 4 3 9. Multiply√√√ and 14/6, together. Ans. 42 V 27° 10. Multiply (+5√ by (√ +2 43 13 Division of Radicals of any Degree. 42. 157. We will suppose, as in the last article, that the radicals have been reduced to equivalent ones having a common index. n Let a and c represent any two radicals of the 1. degree. The quotient of the first by the second may be written, Hence, to divide one radical by another, we have the fol lowing RULE. I. Reduce the radicals to equivalent ones having a common index, II. Divide the co-efficient of the dividend by that of the divisor for a new co-efficient; after this write the radical sign with the common index, and place under it the quotient obtained by dividing the quantity under the radical sign in the dividend by that in the divisor; the result will be the quotient required. EXAMPLES. 1. What is the quotient of c 3/a2b2+b divided by de 3 3 a2 - 86 b2 = d = 2. Divide 23×3/4 by 1/2 × 3/3. n 158. Let a represent any radical of the nth degree. Then we may raise this radical to the mth power, by taking it m times as a factor; thus, But, by the rule for multiplication, this continued product is equal to amn/bm; whence, We have then, to raise a radical to any power, the following RULE. Raise the co-efficient to the required power for a new co-efficient; after this write the radical sign with its primitive index, placing under it the required power of the quantity under the radical sign in the given expression; the result will be the power required. 4 EXAMPLES. 1. (4a3)2 = √(4a3)2 = 1/16a6 2a4a2 = 2a Va. = 486a3/4a2. 2. (33/2u)5=35 3/(2a)5 = 2433/32a5 = 486a 3 : When the index of the radical is a multiple of the exponent of the power to which it is to be raised, the result can be simplified. For, 2a = 2a (Art. 152): hence, in order to square √√√a (Art. we have only to omit the first radical sign, which gives 6 (1/2a)2 = √2a. Again, to square 36, we have 6/36 = we have 36=√√3/36: hence, (36)2=3/36; hence, When the index of the radical is divisible by the exponent of the power to which it is to be raised, perform the division, leaving the quantity under the radical sign unchanged. Extraction of Roots of Radicals of any Degree. 159. By extracting the mth root of both members of equa tion (1), of the preceding article, we find, Whence we see, that to extract any root of a radical of any degree, we have the following RULE. Extract the required root of the co-efficient for a new co-efficient; after this write the radical sign with its primitive index, under which place the required root of the quantity under the radical sign in the given expression; the result will be the root required. 159*. If, however, the required root of the quantity under the radical sign cannot be exactly found, we may proceed in the following manner. If it be required to find the m2 root of ed, the operation may be indicated thus, Consequently, when we cannot extract the required root of the quantity under the radical sign, Extract the required root of the co-efficient for a new co-efficient; after this, write the radical sign, with an index equal to the product of its primitive index by the index of the required root, leaving the quantity under the radical sign unchanged. When the quantity under the radical is a perfect power, of the degree of either of the roots to be extracted, the result can be simplified. 8u3= 3/8a3 = 1/2a. 5 In like manner, √ √ 9a2 = √√ √ 9a2 = √3a. Different Roots of the same Power. 160. The rules just demonstrated depend upon the principle, that if two quantities are equal, the like roots of those quantities are also equal. This principle is true so long as we regard the term root in its general sense, but when the term is used in a restricted sense, it requires some modification. This modification is parti cularly necessary in operating upon imaginary expressions, which are not roots, strictly speaking, but mere indications of operations which it is impossible to perform. Before pointing out these modifications, it will be shown, that every quantity has more than one cube root, fourth root, &c. It has already been shown, that every quantity has two square roots, equal, with contrary signs. |