Ex. 2. If a = 4, b = 9, x = 6, find the value of 8 bx2 27 a3 13. If one factor of a product is equal to 0, the product must be equal to 0, whatever values the other factors may have. A factor 0 is usually called a zero factor. For instance, if x = 0, then ab3xy contains a zero factor. Therefore ab3xy2 = 0 when x = 0, whatever be the values of a, b, y. Again, if c = 0, then c3 = 0; therefore ab2c3 = 0, whatever values a and b may have. 14. DEFINITION. The square root of any proposed expression is that quantity whose square, or second power, is equal to the given expression. Thus the square root of 81 is 9, because 92 = 81. The square root of a is denoted by /a, or more simply a. Similarly the cube, fourth, fifth, etc., root of any expression is that quantity whose third, fourth, fifth, etc., power is equal to the given expression. The roots are denoted by the symbols V, V, V, etc. 3/27 = 3; because 33 = 27. √32 = 2; because 25 = 32. The symbol is sometimes called the radical sign. = 8. Ex. 1. Find the value of 5√(6 a3b1c), when a = 3, b = 1, c = = 5 × √(6 x 27 x 8) = 5 x √1296 If a = 8, c = 0, k = 9, x = 4, y = 1, find the value of If a = 4, b = 1, c = 2, d = 9, x = 5, y = 8, find the value of 15. We now proceed to find the numerical value of expressions which contain more than one term. In these, each term can be dealt with singly by the rules already given, and by combining the terms the numerical value of the whole expression is obtained. 16. We have already, in Art. 8, called attention to the importance of carefully distinguishing between coefficient and index; confusion between these is such a fruitful source of error with beginners that it may not be unnecessary once more to dwell on the distinction. Ex. 1. When c = 5, find the value of c ·4c+2c83c2. 625; 53 = 2 × 5 × 5 × 5 = 250; 17. By Art. 13 any term which contains a zero factor is itself zero, and may be called a zero term. Ex. 1. If a = 2, b = 0, x = 3, y = 1, find the value of NOTE. The two zero terms do not affect the result. Ex. 2. Find the value of x2 - a2y + 7 abx y3, when a2y + 7 abx -- § y3 = § × 72 — 52 × 1 + 0 − § × 13 =29-25-2} = 11%. NOTE. The zero term does not affect the result. 18. In working examples the student should pay atten tion to the following hints: 1. Too much importance cannot be attached to neatness of style and arrangement. The beginner should remember that neatness is in itself conducive to accuracy. 2. The sign should never be used except to connect quantities which are equal. Beginners should be particularly careful not to employ the sign of equality in any vague and inexact sense. 3. Unless the expressions are very short the signs of equality in the steps of the work should be placed one under the other. 4. It should be clearly brought out how each step follows from the one before it; for this purpose it will sometimes be advisable to add short verbal explanations; the importance of this will be seen later. EXAMPLES I. d. If a = 2, b = 3, c = 1, d= 0, find the numerical value of 1. 6a5b-8c+9 d. 2. 3a-4b+ 6c+ 5 d. 3. 6ab-3 cd +2da-5cb+2db. 4. abcbcd + cda + dab. 5. 3 abc-2 bcd+2 cda-4 dab. If a = 1, b = 2, c = 3, d= 0, find the numerical value of 11. a3 + b3 + c3 + d3. 13. 3 abc b2c - 6 a3. 12. bc3 — a3 — b3 — § ab3c. 14. 2 a2+2 b2 + 2 c2 + 2 d2-2 bc - 2 cd-2 da 2 ab. 15. a2 + 2 b2 + 2 c2 + d2 + 2 ab + 2 bc + cd. 16. 2 c2+2 a2+ 2 b2 - 4 cb + 6 abcd. 17. 13 a2+1c4 + 20 ab 11 16 ac - 16 bc. CHAPTER II. NEGATIVE QUANTITIES. ADDITION OF LIKE TERMS 1 19. In his arithmetical work the student has been accustomed to deal with numerical quantities connected by the signs and; and in finding the value of an expression such as 14+7/ 31 +6 — 4 he understands that the quantities to which the sign + is prefixed are additive, and those to which the sign is prefixed are subtractive, while the first quantity, 13, to which no sign is prefixed, is counted among the additive terms. The same notions prevail in Algebra; thus in using the expression 7a+3b-4c-2d we understand the symbols 7 a and 3b to be additive, while 4c and 2 d are subtractive. 20. But in Arithmetic the sum of the additive terms is always greater than the sum of the subtractive terms; and if the reverse were the case, the result would have no arithmetical meaning. In Algebra, however, not only may the sum of the subtractive terms exceed that of the additive, but a subtractive term may stand alone, and yet have a meaning quite intelligible. Hence all algebraic quantities may be divided into positive quantities and negative quantities, according as they are expressed with the sign + or the sign —; and this is quite irrespective of any actual process of addition and subtraction. This idea may be made clearer by one or two simple illustrations. (i) Suppose a man were to gain $100 and then lose $70, his total gain would be $30. But if he first gains $70 and then loses $100 the result of his trading is a loss of $30. |