a2 xa is a 2, the first numerator. (Art. 146.) The fractions reduced are therefore to lower terms. 2a8 5a5 2a3 Ans. and or 5a7 5a7, 5a2 5a2 (Art. 145.) 5 and 3x2 dx 7. Multiply into 3dx3 3d 4x3 SECTION IX. EVOLUTION AND RADICAL QUANTITIES:* ART. 240. IF a quantity is multiplied into itself, the product is a power. On the contrary, if a quanis resolved into any number of equal factors, each of these is a root of that quantity. Thus b is the root of bbb; because bbb may be resolved into the three equal factors b, and b, and b. In subtraction, a quantity is resolved into two parts. In division, a quantity is resolved into two factors. In evolution, a quantity is resolved into equal factors. 241. Aroot of a quantity, then, is a factor which multiplied into itself a certain number of times will produce that quantity. The number of times the root must be taken as a factor, to produce the given quantity, is denoted by the name of the root. Thus 2 is the 4th root of 16; because 2×2×2×2=16, where 2 is taken four times as a factor, to produce 16. So a is the square root of a; for a3 × a3 =a*. (Art.233.) And a is the cube root of a; for a2 Xaa Xaa=a®. And a is the 6th root of a; for a×a×a×a×a×a=a®. Powers and roots are correlative terms. If one quantity is a power of another, the latter is a root of the former. As 6 is the cube of b; b is the cube root of b2. As 9 is the square of 3; 3 is the square root of 9. 242. There are two methods in use, for expressing the roots of quantities, one by means of the radical sign, and the other by a fractional index. The latter is generally to be preferred. But the former has its uses on particular oc casions. When a root is expressed by the radical sign, the sign is placed over the given quantity, in this manner a. *Newton's Arithmetic, Maclaurin, Emerson, Euler, Saunderson, and Simpson. 2 Thus *√a is the 2d or square root of a And 3√a is the 3d or cube root. "Va+y is the nth root of a+y. 243. The figure placed over the radical sign, denotes the number of factors into which the given quantity is resolved; in other words, the number of times the root must be taken as a factor, to produce the given quantity. So that vax %√a=a. And 1 3 3√ax 3 vax3 √α=α. And "Vax "√a....n times =a. The figure for the square root is commonly omitted; √ being put for Va. Whenever, therefore, the radical sign is used without a figure, the square root is to be understood. 244. When a figure or letter is prefixed to the radical sign, without any character between them; the two quantities are to be considered as multiplied together. Thus 2 va, is 2xva, that is, 2 multiplied into the root of a, or which is the same thing, twice the root of a. And xvb, is xx√b, or x times the root of b. When no co-efficient is prefixed to the radical sign, 1 is always to be understood; a being the same as 1 Va, that is, once the root of a. 245. The method of expressing roots by radical signs, has no very apparent connection with the other parts of the scheme of algebraic notation. But the plan of indicating them by fractional indices, is derived directly from the mode of expressing powers by integral indices. To explain this, let a be a given quantity. If the index be divided into any number of equal parts, each of these will be the index of a root of aR. 1 Thus the square root of a, is a3. For, according to the definition, (Art. 241.) the square root of a is a factor, which multiplied into itself will produce a. But a3× a3 =a®. (Art. 233.) Therefore, as is the square root of a The index of the given quantity a, is here divided into the two equal parts 3 and 3. Of course, the quantity itself is resolved into the two equal factors a3 and a3. The cube root of a is a2. For a2 xa2x a2 =α... Here the index is divided into three equal parts, and the quantity itself resolved into three equal factors. The square root of a2 is c1 or a. For axa=a*.. By extending the same plan of notation, fractional indides are obtained. Thus, in taking the square root of a1 or a, the index 1 is divided into the two equal parts and ; and the root is a On the same princple, The cube root of a, is a3=3√a The fourth root is a*=*√a The nth root, is a="/a, &c. And the nth root of a+x, is (a+x)" = "√a+x. 1 246. In all these cases, the denominator of the fractional index, expresses the number of factors into which the given quantity is resolved. 247. It follows from this plan of notation, that axa} = a a3=a+, a3× a3× a3± a3+3+}=a1, &c. παι, where the multiplication is performed in the same manner, as the multiplication of powers, (Art. 233,) that is, by adding the indices. 248. Every root as well as every power of 1 is 1. (Art. 209.) For a root is a factor which multiplied into itself will produce the given quantity. But no factor except 1 can produce 1, by being multiplied into itself. So that 1", 1, VI, "/1, &c. are all equal. 249. Negative indices are used in the notation of roots, as well as of powers. See art. 207. |