10. Each of the letters composing a term is called a dimension of the term, and the number of letters involved is called the degree of the term. Thus the product abc is said to be of three dimensions, or of the third degree; and art is said to be of five dimensions, or of the fifth degree. A numerical coefficient is not counted. Thus 8a2b5 and a2b5 are each of seven dimensions, or of the seventh degree. 11. But it is sometimes useful to speak of the dimensions of an expression with regard to any one of the letters it involves. For instance, the expression 8a34c, which is of eight dimensions, may be said to be of three dimensions in a, of four dimensions in b, and of one dimension in c. 12. A compound expression is said to be homogeneous when all its terms are of the same degree. Thus 8a6-a+b2+9ab5 is a homogeneous expression of six dimensions, or of the sixth degree. 13. In dealing with algebraical expressions, where the letters denote numerical quantities, we may make use of the principles with which the student is familiar in Arithmetic. Thus ab and ba each denote the product of the two quantities represented by the letters a and b, and have therefore the same value. Again, the expressions abc, acb, bac, bca, cab, cba have the same value, each denoting the product of the three quantities a, b, c. It is immaterial in what order the factors of a product are written; it is usual, however, to arrange them in alphabetical order. Example 1. If x=5, y=3, find the value of 4x2y3. 4x2y3=4 x 52 x 33 = 4 x 25 x 27 = 2700. Example 2. If a=4, b=9, x=6, find the value of 8bx2 27a3 14. If one factor of a product is equal to 0, the whole product must be equal to 0, whatever values the other factors may have. A factor 0 is sometimes called a 66 zero factor." Example 1. If x=0, then ab3xy2=0, whatever be the values of a, b, y. Example 2. If c=0, then ab2c3=0, whatever values a, b may have. EXAMPLES I. b. If a=7, b=2, c=0, x=5, y=3, find the value of 1. 2. a3b. 3. 8b2y. 4. 3xy2. 11. 4ax2. 6. by2. 7. xy1. 8. a3c. 9. a2cy. 10. 8x3y. ab3x. 12. x2y. 5. 362x. If a=2, b=3, c=1, p=0, q=4, r=6, find the value of 15. 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, &c., root of any expression is that quantity whose third, fourth, fifth, &c., power is equal to the given expression. The roots are denoted by the symbols,,, Examples. 27=3; because 33=27. /32=2; because 25=32. &c. The symbol is sometimes called the radical sign. Example 1. Find the value of 5√(6a3b1c), when a=3, b=1, c=8. 5 √(6a3b*c)=5 × √(6 × 33 × 14 × 8) =5× √(6 × 27 × 8) =5× √/1296 = 5 x 36 = 180. If a=8, c=0, k=9, x=4, y=1, find the value of 21. √(8ac). If a=4, b=1, c=2, d=9, x=5, y=8, find the value of 16. In working examples the student should pay attention to the following hints: 1. It should be clearly brought out how each step follows from the one before it; for this purpose short verbal explanations are often necessary. 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 several steps of the work should be placed one under the other. 4. In all work too much importance cannot be attached to neatness of style and arrangement. The beginner should remember that neatness is in itself conducive to accuracy. Example 1. Find the value of fab − 7x2 – ‡ay2+263, when a=5, b=4, x=3, y=2. ab-7x2 - Lay2+263=1.5.4-7.32-.5.22+2.43 Example 2. =6-63-45 +128 =26. Find the value of ‡x2 - a2y+7abx - y3, when a=5, b=0, x=7, y=1. x2- a2y+7abx - §y3= §. 72 – 52.1+0 - §. 13 =203-25-21 NOTE. In the last example the zero term does not affect the result. Example 3. When p=9, r=6, k=4, find the value of |