11. The square root of any proposed quantity is that quantity whose square, or second power, gives the proposed quantity. The cube root, is that quantity whose cube gives the proposed quantity, &c. The signs √, or †, V, V, &c. are used to express the square, cube, biquadrate, &c. roots of the quantities before which they are placed. 7 3 These roots are all represented by the fractions,,, &c. placed a little above the quantities, to the right. Thus a, a, a, a, represent the square, cube, fourth and nth root of a, respectively; a, a, a, represent the square root of the fifth power, the cube root of the seventh power, the fifth root of the cube of a. n' 12. If these roots cannot be exactly determined, the quantities are called irrational or surds. 13. Points are made use of to denote proportion, thus a b c d, signifies that a bears the same proportion to b that e bears to d. 14. The number prefixed to any quantity, and which shews how often it is to be taken, is called its coefficient. Thus, in the quantities 7ax, 6by, and 3 dx, 7, 6, and 3 are called the coefficients of ax, by, and de respectively. When no number is prefixed, the quantity is to be taken once, or the coefficient 1 is understood. These numbers are sometimes represented by letters, which are called coefficients. 15. Similar, or like algebraical quantities are such as differ only in their coefficients; 4a, 6ab, 9a2, 3a2bc, are respectively similar to 15a, 3ab, 12 a2, 15a bc, &c. Unlike quantities are different combinations of letters; thus, ab, a2b, ab2, abc, &c. are unlike. 16. A quantity is said to be a multiple of another, when it contains it a certain number of times exactly thus 16a is a multiple of 4a, as it contains it exactly four times. : 17. A quantity is called a measure of another, when the former is contained in the latter a certain number of times exactly; thus, 4a is a measure of 16a. 18. When two numbers have no common measure but unity, they are said to be prime to each other. 19. A simple algebraical quantity is one which consists of a single term, as a2bc. 20. A binomial is a quantity consisting of two terms, as a + b, or 2a − 3bx. A trinomial is a quantity consisting of three terms, as 2a + bd + 3c. 21. The following examples will serve to illustrate the method of representing quantities algebraically: : Let a = 8, b = 7, c = 6, d · bd = 56+ 6 3b + 5 20 8+7 d 9 3 14 +24 35= 21 12 H + 6 1 3 ce2 + d3 = 25 × 2 - 18 + 125 18 + 125 = 157. *(2.) To add and subtract simple Algebraical 22. The addition of algebraical quantities is performed by connecting those that are unlike with their proper signs, and collecting those that are similar into one sum. 23. -ax by cy Sum 4ax+by - cy Sum 2a + n − 1b Subtraction, or the taking away of one quantity from another, is performed by changing the sign of the quantity to be subtracted, and then adding it to the other by the rules laid down in *(3.) To multiply simple Algebraical Quantities. 24. The multiplication of simple algebraical quantities must be represented according to the notation pointed out in Art. 4 and 5. Thus, a × b, or a b, represents the product of a multiplied by b; abc, the product of the three quantities a, b, and c. It is also indifferent in what order they are placed, ax b and bx a being equal. 25. If the quantities to be multiplied have coefficients, these must be multiplied together as in common arithmetic; the literal product being determined by the preceding rules. Thus, 3a x 5b 15ab; because = 3 x ax 5 × b = 3 × 5 × a x b = 15 ab. 26. The powers of the same quantity are multiplied together by adding the indices: thus, a x a3 = a3 ; for a ax aaaaaaaa. In the same manner, n a” × a” = a”+”; and 3a2x3 × 5 a x y2 27. If the multiplier or multiplicand consist of several terms, each term of the latter must be multiplied by every term of the former, and the sum of all the products taken, for the whole product of the two quantities. * (4.) To divide simple Algebraical Quantities. 28. To divide one quantity by another, is to determine how often the latter is contained in the former, or what quantity multiplied by the latter will produce the former. Thus, to divide ab by a is to determine how often a must be taken to make up ab; that is, what quantity multiplied by a will give ab; which we know is b. From this consideration are derived all the rules for the division of algebraical quantities. If only a part of the product which forms the divisor be contained in the dividend, the division must be represented according to the direction in Art. 6, and the quantities contained both in the divisor and dividend expunged. Thus 15a2bc divided by 3a2bx is 5bc 15a2bc which 3a2bx is equal to ; expunging from the dividend and from the divisor the quantities 3, a3, and b. *(5.) To reduce Fractions to others of equal value which have a common denominator. 29. Fractions are changed to others of equal value with a common denominator, by multiplying each numerator by every denominator except its own, for the new numerator; and all the denominators together for the common denominator. the numerator and denominator of each fraction having been multiplied by the same quantity viz.—the product of the denominators of all the other fractions. 30. When the denominators of the proposed fractions are not prime to each other, find their greatest common measure; multiply both the numerator and |