...principle, which is merely an extension of that already proved in the preceding Article. The nth root of the product of any number of factors is equal to the product of the nth roots of those factors. Or in algebraic language For raise each of these expressions to the...

...quantities themselves are equal : hence, y abc . . . . = ya X y SX ус . . . that is, íAe nth root of the product of any number of factors, is equal to the product of their nth roots. 1. Let us apply the above principle in reducing to its sim plest form the imperfect power,...

...following principle, which is merely an extension of that already proved in (Art. 51). The n"1 root of the product of any number of factors is equal to the product oftheri* roots of the different factors. Or, in algebraic language, = V~a XV * XV~c X Raise each of...

...hence the logarithm of a negative number is imaginary. Properties of Logarithms. (1) The logarithm of the product of any number of factors is equal to the sum of the logarithms of those factors. Let p, Q, к, s be any quantities, and x, у, z, v their logarithms...

...square root of 365, we decompose 365 into the prime factors 5 and 73. Now, it is obvious that the root of the product of any number of factors is equal to the product of their roots, hence, the square root of 365 is equal to the square root of 5 x 73, which is equal to the square...

...n'* powers are equal, the quantities themselves are equal : hence, XX that is, the nth roo< o/" Me product of any number of factors, is equal to the product of their nth roots. 1. Let us apply the above principle in reducing to its sim plest form the imperfect power,...

...g- y^/', & c -; = lo g- y + lo g- y' + :1 °g- y"» &e From which it is evident, that the logarithm of the product of any number of factors is equal to the sum of the logarithms of those factors. Hence, if all the factors of a given number, in any case of...

...of computation. Tin following properties of logarithms are common to all systems : 1 . The logarithm of the product of any number of factors, is equal to the sum of the logarithms of the factors, taken separately. 2. The logarithm of the quotient of one number...

...(NX N' X N") = x+x+x" = log N -f- bg N'+ log N". Hence we come to the conclusion that the logarithm of the product of any number of factors is equal to the sum of the logarithms of the factors (THEOREM 1.) Now if we take the logarithms of 2 and 3 as hereafter...

..." Surds of any degree may be simplified by application of the following principle : " " The n* root of the product of any number of factors is equal to the product of the n'1" roots of those factors." In Day's Algebra, edition- of 1848, under the head of Multiplication...