40. Rule. To multiply two simple expressions together, multiply the coefficients together and prefix their product to the product of the different letters, giving to each letter an index equal to the sum of the indices that letter has in the separate factors.
The rule may be extended to cases where more than two expressions are to be multiplied together.
Ex. 1. Find the product of x^2, x^3, and x8.
The product == x^2 x^3 x^8 = x^2+3 x^8 = x^2+3+8 = x^13.
The product of three or more expressions is called the continued product.
Ex. 2. Find the continued product of 5x2y3, 8y2z5, and 3xz4.
The product = 5 x2y3 x 8y2z5 x 3xzr = 120 x3y5z9,
MULTIPLICATION OF A COMPOUND EXPRESSION BY A SIMPLE EXPRESSION.
41. By definition,
(a+ b)m= m+m+m-+--- taken a+ b times =(m+m-+m + --- taken a times),
together with (m+ m +m + --- taken b times) =am ee le
Also (a-b)m= m+m-+m-+--- taken a -b times =(m+m-+m +--- taken times),
diminished by (m+m-+m-+ --- taken b times)
Similarly, (a-b+c)m = am - bm + cm.
Hence the product of a compound expression by a single factor is the algebraic sum of the partial products of each term of the compound expression by that factor. This is known as the Distributive Law for Multiplication.
