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In multiplication of algebraic quantities there is one general rule for the signs ; namely, when the signs of the factors are both affirmative or both negative, the product is affirmative ; but if one of the factors be affirmative and the other negative, then the product is negative.*
in the prod
* That like signs make +, and unlike signs uct, may be shewd thus :
When both the factors are simple quantities.
Multiply the coefficients of the two terms together, to the product annex all the letters of the terms, and prefix the proper sign.
1. When ta is to be multipled by +b; it implies, that ta is to be taken as many times, as there are units in b; and since the sum of any number of affirmative terms is affirmative, it follows, that fax +b makes t-ab.
2. When two quantities are to be multiplied together; the result will be exactly the same, in whatever order they are placed ; for a times b is the same as b times a ; and therefore, when a is to be multiplied by +b, or + b by -a, it is the same thing as taking as many times as there are units in +b; and since the sum of any number of negative terms is negative, it follows, that -ax+b, or tax-b; makes or produces –ab.
a is to be multiplied by b; here a is to be subtracted as often as there are units in b; but subtracting negatives is the same as adding affirmatives, by the demonstration of the rule for subtraction; consequently the quotient is 6 times a, or tab.
NOTE 1. To multiply any power by another of the same root add the exponent of the multiplier to that of the multiplicand, and the sum will be the exponent of their product.
Thus the product of as multiplied into a3 iş a5+3, or a8.
Again, the product of atx multiplied into a+x.is at***. And that of tyl" into xtsi is xtyl
Otherwise. Since a-a=, therefore a-ax-b is also = 0, beause o multiplied by any quantity is still o ; and since the first term of the product, or ax-b, Şab, by the second case
there. fore the last term of the product, or -ax-b, must be tab, tò make the sum =0, or --ab-tab=0 ; that is, -ax
This rule is equally applicable, where the exponents of any roots of the same quantity are fractional.
- 를 Thus, the product of a' multiplied into a' is a Xa = *+=+=a=a. In like manner, ** Xx Xx't
=++} = * =*
Hence it appears, that, if a surd square root be multiplied into itself, the product will be rational ; and if a 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.
5+3 Thus, a* x axata +
우 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 6+1==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.
Compound surds of the same quantity are multiplied in the same manner as simple ones.
공 Thus, at a7* x 27 x atx]* eta] = at*; 03#**# * a ****** = a* t**j* =2*+*”.
So likewise Vatex x Va+* =V atx=a+
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. Thus, vix_3=V7X3=V21: vixiz 70 29 =
=> 14cxy .
It may not be improper to observe, that unequal surds have sometimes a rational product.