Here 4 times the negative index being —8, and S to be carried, the difference —5 is the index of the product. 4. To raise 1*0045 to the 365th root. Divide the logarithm of the given number by the index of the power, and the number answering to the quotient will be the root required. Note. When the index of the logarithm is negative, and cannot be divided by the divisor without a remainder, increase the index by a number, that will render it exactly divisible, and carry the units borrowed, as so many tens, to the first decimal place; and divide the rest as usual. Here the divisor 2 is contained exactly once in the nega tive index -2, and therefore the index of the quotient is —1. 6. To find the third root of '00048. Here the divisor 3 not being exactly contained in 4, 4 is augmented by 2, to make up 6, in which the divisor is contained just 2 times; then the 2, thus borrowed, being carried to the decimal figure 6, makes 26, which, divided by 3, gives 8, &c. For-4--6+2. ALGEBRA, DEFINITIONS And NOTATION.. 1. ALGEBRA is the art of computing by symbols. It is sometimes also called Analysis; and is a general kind of arithmetic, or universal way of computation. 2. In Algebra, the given, or known quantities are usually denoted by the first letters of the alphabet, as a, b, c, d, &c. and the unknown, or required quantities, by the last letters, as *, y, z. Note. The signs, or characters, explained at the beginning of Arithmetic, have the same signification in Algebra. 3. Those quantities, before which the sign+ is placed, are called positive, or affirmative; and those, before which the sign is placed, negative. And it is to be observed, that the sign of a negative quantity is never omitted, nor the sign of an affirmative one, except it be a single quantity, or the first in a series of quantities, then the sign + is frequently omitted: thus a signifies the same as +a, and the series a+h—c+d the same as +a+b —c+dj so that, if any single quantity, or if the first term in any number of terms, have not a sign before it, then it is always understood to be affirmative. 4. Like signs are either all positive, or all negative; but signs are unlike, when some are positive and others negative. 5. Single, or simple quantities consist of one term only, as a, b, x. In multiplying simple quantities, we frequently omit the sign X, and join the letters; that, ab signifies the same as axb; and abc, the same as axbxc. And these products, viz. axb, or ab, and abc, are called single or simple quantities, as well as the factors, viz. a, b, c, from which they are produced, and the same is to be observed of the products, arising from the multiplication of any number of simple quantities. 6. If an algebraical quantity consist of two terms, it iS called a binomial, as a+b; if of three terms, a trinomial, as a+b+c; and if of four terms, a quadrinomial, as a+b+c+d; and if there be more terms, it is called a multinomial, or polynomial; all of which are compound quantities. When a compound quantity is to be expressed as multiplied by a simple one, then we place the sign of multiplication between them, and draw a line over the compound quantity only; but when compound quantities are to be represented as multiplied together, then we draw a line over each of them, and connect them with a proper sign. Thus, a+oxc denotes, that the compound quantity a+b is multiplied by the simple quantity c; so that if a were 10, b 6, and c 4, then would abxc be 10+6x4, or 15 into 4, which is 64; and a+bxc+J expresses the product of the compound quantities a+b and c+d multiplied together. 7. When we would express, that one quantity, as d, is greater than another, as b, we write a b, or a b; and if we would express, that a is less than b, we write ab, or a b. 8. When we would express the difference between two quantities, as a and b, while it is unknown/which is the greater of the two, we write them thus, a n b, which denotes the difference of a and b. 9. Powers of the same quantities or factors are the products of their multiplication: thus axa, or aa, denotes the square, or second power of the quantity, represented a; axa Xa, or aaa, expresses the cube, or third power; and axa a Xo, or aaaa, denotes the biquadrate, or fourth power of a, &c. And it is to be observed, that the quantity a is the root of all these powers. Suppose a=5, then will aaaa=5X5= 25= the square of 5; aaa=aXaXa=5x5XS=V2S= the cube Vol. I. Ff of 5 and aaaa^axaxaxa—5X5x5x5—625= the fourth power of 5. 10. Powers are likewise represented by placing above the root, to the right, a figure expressing the number of factors, that produce them. Thus, instead of aa, we write a2; instead of aaa, we write a3; instead of aaaa, we write a*, &c' 11. These figures, which express the number of factors* that produce powers, are called their indices, or exponents; thus, 2 is the index or exponennt of a2; 3 is that of x3; 4 is that of x, &c. But the exponent of the first power, though generally omitted, is unity, or 1; thus a1 signifies the same as a, namely, the first power of a; axa, the same as a1xal, or a1*1, that is, a3, and a*xa is the same as a2 xa», or a3+1, or a3. 12. In expressing powers of compound quantities, we usually draw a line over the given quantity, and at the end of the line place the exponent of the power. Thus, 2 a+ denotes the square or second power of a+b, consider 3 ed as one quantity; a+ the third power; a+6\ *the fourth power, &c* And it may be observed, that the quantity a+by called the first power of a+b, is the root of all these powers. Let a=4 and b=2, then will a+b become 4+2, or 6; and a+b]*=4+2]=6*=6x6=36, the square of 6; also a+6| 3 =4+2 =63=6X6X6=216, the cube of 6. 13. The division of algebraic quantities is Very frequent ly expressed by writing down the divisor under the dividend with a line between them, in the manner of a vulgar fraction : thus, represents the quantity arising by dividing a by c; c a с 144 so that if a be 144 and c 4, then will be or 36. a+b And denotes the quantity arising by dividing a+b by |