pression b3, which equals b× b× b, the 3 is the index or exponent of the power to which b is to be raised, and it indicates that b is to be used as a factor 3 times. An exponent, like a coefficient, may be numerical, literal, or mixed; thus, x, x, xn, &c. b If no exponent is written, a unit is understood; thus b1, a = a1, &c. Coefficients and Exponents must be carefully distinguished from each other. A Coefficient shows the number of times a quantity is taken to make up a given sum; an Exponent shows how many times a quantity is taken as a factor to make up a given product; thus 4 x = x + x + x + x, and x = x x x x x x x. 25. The product obtained by taking a quantity as a factor a given number of times is called a power, and the exponent shows the number of times the quantity is taken. 26. A Roor of any quantity is a quantity which, taken as a factor a given number of times, will produce the given quantity. A RooT is indicated by the radical sign, , or by a fractional exponent. When the radical sign, ✔, is used, the index of the root is written at the top of the sign, though the index denoting the second or square root is generally omitted; thus, √, or x, means the second root of x; Every quantity is considered to be both the first power and the first root of itself. 27. The RECIPROCAL of a quantity is a that quantity. Thus, the reciprocal of 5 is 28. A MONOMIAL is a single term; as a, or 3x, or 5 bxy. 29. A POLYNOMIAL is a number of terms connected with each other by the signs plus or minus; as x + y, or 3 a + 4x 7aby. 30. A BINOMIAL is a polynomial of two terms; as 3x+3y, or x — y. 31. A RESIDUAL is a binomial in which the two terms are connected by the minus sign, as x y. 32. SIMILAR TERMS are those which have the same powers of the same letters, as x and 3x, or 5 ax3 and 2 ах3. But x and x2, or 5 a and 5 b, are dissimilar. 33. The DEGREE of a term is denoted by the sum of the exponents of all the literal factors. Thus, 2 a is of the first degree; 3 a2 and 4 ab are of the second degree; and 6 a3 x is of the seventh degree. 34. HOMOGENEOUS TERMS are those of the same degree. Thus, 4 a2 x, 3 abc, x2y, are homogeneous with each other. 35. To find the numerical value of an algebraic expression when the literal quantities are known, we must substitute the given values for the letters, and perform the operations indicated by the signs. The numerical value of 7 a bc2 when a = 4, Find the numerical values of the following expressions, when a 2, b = 13, c = 4, d = 15, m = = 5, and n = 7. 36. Write in algebraic form: 1. The sum of a and b minus the difference of m and n. (m>n.) 2. Four times the square root of the sum of a, b, and c. 3. Six times the product of the sum and difference of c and d. (c> d.) 4. Five times the cube root of the sum of a, m, and n. SECTION IV. ADDITION. 37. ADDITION in Algebra is the process of finding the aggregate or sum of several quantities. For convenience, the subject is presented under three cases. CASE I. 38. When the terms are similar and have like signs. 1. Charles has 6 apples, James 4 apples, and William 5 apples; how many apples have they all? Therefore, when the terms are similar and have like signs: RULE. Add the coefficients, and to their sum annex the common letter or letters, and prefix the common sign. 8. What is the sum of a x2, 3 ax2, 2 a x2, and 4 ax2? 9. What is the sum of 3 bx, 4bx, 6bx, and bx?146 10. What is the sum of 2xy, 6xy, 10 xy, and 8xy? 26 39. When the terms are similar and have unlike signs. 1. A man earns 7 dollars one week, and the next week earns nothing and spends 4 dollars, and the next week earns 6 dollars, and the fourth week earns nothing and spends 3 dollars; how much money has he left at the end of the fourth week? If what he earns is indicated by +, then what he spends will be indicated by, and the example will appear as follows: Therefore, when the terms are similar, and have unlike signs: |