and the 2, 3, and 4 are exponents. The expressions are read, a square,' a cube" or "a third," a fourth;" and if we have a", in which a enters m times as a factor, it is read “a to the mth," or simply "a mth." The exponent 1 is generally omitted. Thus, a1 is the same as a, each denoting that a enters but once as a factor. 15. A power is a product which arises from the multiplication of equal factors. Thus, the square, or second of power, a. the cube, or third power, of a. = the fourth power of a. axa = = a2 = a xaxa = a3 = axaxaxa = = a1 = ахах = am the mth power of a. 16. A root of a quantity is one of the equal factors. The radical sign (√ ̄ ̄ ̄ ̄), when placed over a quantity, indicates that a root of that quantity is to be extracted. The root is indicated by a number written over the radical sign, called an index. When the index is 2, it is generally omitted. Thus, Va, or Va, indicates the square root of a. Va indicates the fourth root of a. Wa indicates the mth root of a. 17. An algebraic expression is a quantity written in algebraic language. Thus, 3a is the algebraic expression of three times the number denoted by a; 5a2, that of five times the square of a; 7ab2, that of seven times the cube of a multiplied by the square of b; 3a-56, that of the differ -- ence between three times a and five times b; and 2 a2 3 ab +462, that of twice the square of a, diminished by three times the product of a by b, augmented by four times the square of b. 18. A term is an algebraic expression that can be written without the aid of either of the signs or —. Thus, 3a, 2ab, 5a2b2, are terms. A term may be preceded by either of the signs + or In this case the sign is called the sign of the term, and is used to show the sense in which the term is taken. Thus +3a shows that 3 a is taken positively, and -5ab2 shows that 5a2b2 is taken negatively. 19. The degree of a term is the number of its literal factors. Thus, 3a is a term of the first degree, because it contains but one literal factor; 5a2 is a term of the second degree, because it contains two literal factors (the factors in this case are equal); 7a3b is a term of the fourth degree, because it contains four literal factors. The degree of a term is determined by the sum of the exponents of all its letters. 20. A monomial is a single term, unconnected with any other by the signs + or. Thus, 3a2, 363 a, are monomials. 21. A polynomial is a collection of terms connected by the signs or; as, 3a-5, or 2 a3 — 3b+ 4 b2. 22. A binomial is a polynomial of two terms; as, a+b, 3a2 - c2, 6ab — c2. 23. A trinomial is a polynomial of three terms; as, abc — a3 + c3, ab — gh—f. 24. Homogeneous terms are those which are of the same degree. Thus, the terms abc, a3, + c3, are homogeneous; as are the terms ab, — gh. 25. A polynomial is homogeneous when all its terms are homogeneous. Thus, the polynomial abc a3+c3 is homogeneous, but the polynomial ab — gh —ƒ is not homogeneous. 26. Similar terms are those which have a common unit; that is, have a common literal part. Thus, 7 ab+3ab -2ab are similar terms, and so also are 4a2b2 — 2a2b2 — 3 a2b2 ; but the terms of the first polynomial and of the last are not similar. 27. The vinculum,; the bar, ; the parentheses, (); and the brackets, [], are each used to connect two or more terms which are to be operated upon as a whole. of the expressions a x Thus, each a+b+cxx, +b (a+b+c) xx, [a+b+c] × x, +c indicates that the sum of a, b, and c, is to be multiplied by x. 28. The reciprocal of a quantity is 1 divided by that quantity. Thus, 29. The numerical value of an algebraic expression is the result obtained by assigning a numerical value to each letter, and then performing the operations indicated. Thus, the numerical value of the expression ab+be+d, when a=1, b=2, c=3, and d = 4, is 1×2+2×3+4=12, by performing the indicated operations. EXERCISES IN WRITING ALGEBRAIC EXPRESSIONS. 1. Write a added to b. Ans. a+b. 2. Write b subtracted from a. Ans. a-b. Write the following: 3. Six times the square of a, minus twice the square of b. 4. Six times a multiplied by b, diminished by 5 times c cube multiplied by d. 5. Nine times a, multiplied by c plus d, diminished by 8 times b multiplied by d cube. 6. Five times a minus b, plus 6 times a cube into b cube. 7. Eight times a cube into d fourth, into c fourth, plus cube into d fifth, minus 6 times a into b, into c 9 times square. 8. Fourteen times a plus b, multiplied by a minus b, plus 5 times a, into c plus d. 9. Six times a, into c plus d, minus 5 times b, into a plus c, minus 4 times a cube b square. 10. Write a, multiplied by c plus d, plus ƒ minus g. 11. Write a divided by b+c in three ways. 12. Write a b divided by a+b. 13. Write a polynomial of three terms, of four terms, of five, of six. 14. Write a homogeneous binomial of the first degree, of the second, of the third, fourth, fifth, sixth. 15. Write a homogeneous trinomial of the first degree, with its second and third terms negative; of the second degree; of the third; of the fourth. 16. Write in the same column, on the slate or blackboard, a monomial; a binomial; a trinomial; a polynomial of four terms, of five terms, of six terms, and of seven terms; and all of the same degree. |