This rule is equally applicable, when the exponents of any roots of the same quantity are fractional. Thus, the product of a multiplied into a is a is axa Hence it appears, that, if a surd square root be multiplied into itself, the product will be rational; and if à surd cube root be multiplied into itself, and that product into the same root, the product is rational. And, in general, when the sum of the numerators of the exponents is divisible by the common denominator, without a remainder, the product will be rational. Here the quantity a is reduced to a, by actually dividing 8, the numerator of the exponent, by its denominator 4; and the sum of the exponents, considered merely as vulgar fractions, is 4+4=4=2. When the sum of the numerators and the denominator of the exponents admit of a common divisor greater than unity, then the exponent of the product may always be reduced, like a vulgar fraction, to lower terms, retaining still the same value. Thus, x Compound surds of the same quantity are multiplied in the same manner as simple ones. Thus, a+x|* × a+x] = a+x1a = e+xl' = e+xl2 = a+x ; 8 8 So likewise ✔a+x x✔a+x =√a+x=a+2a. And √a+* ×√a+x=√ «+x=a+x 3. And √a+xx√a+x=a+x. These examples shew the grounds, on which the products of surds become rational. NOTE 2. Different quantities under the same radical sign are multiplied together like rational quantities, only the product, if it do not become rational, must stand under the same radical sign. It may not be improper to observe, that unequal surds have sometimes a rational product. CASE III. When both the factors are compound quantities. RULE. Multiply each term of the multiplicand by each term of the multiplier; then add all the products together, and the sum will be the product required. *In the first example, we multiply a+b, the multiplicand, into , the first term of the multiplier, and the product is a2+ab; then we multiply the multiplicand into b, the second term of the multiplier, and the product is ab+b2. The sum of these two products is a2+2ab+b2, as above, and is the square of a+b. In the first example, the like terms of the product, viz. ab and ab, together make 2ab; but in the second example, the terms +ab and ab, having contrary signs, destroy each other, and the product is e-b2, the difference of the squares of a and b. Hence it appears, that the sum and difference of two quantities, multiplied together, produce the difference of their squares. And by the next following example you may ob. serve, that the square of the difference of two quantities, as a and b, is equal to a2-2ab+b2, the sum of their squares minus twice their product. |