be extracted; also, 83 denotes that 8 is to be squared, which is 64, then the cube root of this 64 to be extracted, which is 4, &c. 11. The coefficient of a quantity is the number or letter prefixed to it, and signifies how often the other is to be taken. Thus, 5x2 signifies that the x2 is to be taken five times; here 5 is the coefficient to the x2, and the small 2 over the x is called the index, and is read, five times r squared. Also, 4 x, or 4x, 4 being the coefficient, and denotes four times the square root of x. N. B. When any quantity stands without a coefficient as x, it is supposed to have 1 for a coefficient, for a being the same as 1.r. 12. Quantities are said to be like which differ only in their coefficients. Thus, 4x and -2x are like quantities, 3a2 and 4a2 are like quantities. But 2a and 3x are unlike quantities, a and x being different. 13. An irrational or surd quantity, is that which has no exact root, and can only be expressed by a radical sign or fractional index; as /3, or 3; or 14: Simple quantities are those which consist of one term only, as 4x, 5y, 4xy, &c.; and compound quantities, are those which consist of several terms, as a+x, 3x-4y, x2+2xy + y2, &c. 15. A vinculum is a parenthesis (), or bar used to collect several quantities into one; as (x+y) xa, or x+yxa, signifies that the compound quantity x+y is to be multiplied by the simple quantity a. pressed thus (x+y)3, or x+y]3; and so on for the 4th, 5th, or any other power. These small figures are called the indices. N. B. A letter or any quantity without a small figure placed over the right hand corner, is supposed to have 1 for an index. Thus a is the same as x1. 9. The Sign denotes the square root of the quantity to which it is prefixed. Thus, 25, signifies that the square root of 25 is to be extracted, or being placed over the right hand corner has the signifies that the square root same effect. Thus, 25 of 25 is to be taken. 10. The fraction prefixed to a quantity, as a3, signifies that the square of the cube root of a is to be taken, or 83 signifies that the cube root of 8 is to be extracted, which is 2, and then the 2 to be squared, which is 4; therefore, 83 is equal to 4. Also, 83 denotes the 5th power of the cube root of 8. Also, 16* denotes the 5th power of the 4th root of 16, &c. Or thus, a3 signifies that a, or the quantity which it denotes, is to be squared, and the cube root of that power to 11. The coefficient of a quantity is the number or letter prefixed to it, and signifies how often the other is to be taken. Thus, 5x2 signifies that the x2 is to be taken five times; here 5 is the coefficient to the x2, and the small 2 over the x is called the index, and is read, five times squared. Also, 4 x, or 4x, 4 being the coefficient, and denotes four times the square root of x. N. B. When any quantity stands without a coefficient as x, it is supposed to have 1 for a coefficient, for a being the same as 1.r. 12. Quantities are said to be like which differ only in their coefficients. Thus, 4x and -2x are like quantities, 3a2 and 4a2 are like quantities. But 2a and 3x are unlike quantities, a and x being different. 13. An irrational or surd quantity, is that which has no exact root, and can only be expressed by a radical sign or fractional index; as /3, or 3; or 14: Simple quantities are those which consist of one term only, as 4x, 5y, 4xy, &c.; and compound quantities, are those which consist of several terms, as ax, 3x-4y, x2+2xy + y2, &c. 15. A vinculum is a parenthesis (_), or bar used to collect several quantities into one; as (x+y) xa, or x+yxa, signifies that the compound quantity x+y is to be multiplied by the simple quantity a. 16. The reciprocal of any quantity or number, is that quantity or number inverted; as the reciprocal of x is; the reciprocal of is; the reciprocal of 5 is or .2, and so on. 17. The Sign ... stands for therefore. ADDITION. ADDITION of Algebra is performed by connecting the quantities together by their proper signs, and may be divided into three cases. CASE I. RULE.-When the quantities are like (Def. 12. ) with the same sign, viz. all + or all -, add the coefficients into one sum, to which adjoin the letters common to each term, and prefix the sign. * The reason of this rule is obvious, for 2a added to 3a, make 5a; and 5a added to 4a, make 9a, and 9a added to 6a, make 15a. It is usual in Addition to begin at the bottom of the left hand column, as in the 4th question, begin with 9xy. 3xy2 2x2y -2x y2 x2+2xy+ y2 6 -- √ax+ X2 I а -Зах- х 2 x2+ xy- a √x+2ax x2 3x24xy+ 2a √ a −3√√x +12 1 * In the first example, -4ax, 3ax, and 3ax, are like quantities, which by case second make 2ax. Also, xy and -2xy, are like quantities, which by case second make-xy; and the remaining quantity 2xy2, must be placed in the line with its proper sign, as above. |