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 ax^4 is said to be of
five dimensions, or of the fifth degree.

A numerical coefficient is not counted. Thus 8a^2b^5 and a^2b^5
are each of seven dimensions.

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 8a^3b^4c, 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 dimensions. Thus 8a^6 - a^4b^2 + 9ab^5
is a homogeneous expression of six dimensions.

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 4x^2y^3.

4x^2y^3 = 4 x 5^2 x 3^3,

= 4 x 25 x 27,

= 2700.

Example 2. If a = 4, b = 9, x = 6, find the value of 8bx^2 / 27a^3.

8bx^2 / 27a^3= 8x9x6^2 / 27 x4^3,

= 8 x 9 x 36 / 27 x 64,

= 3/2.

= 1-1/2.

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 "zero factor."
