POWERS, in arithmetic and algebra, are the products arising from the continual multiplication of a number, or quantity, into itself: thus, 2, 4, 8, 16, 32, &c. are the powers of the number 2 ; and a, a', frt, &c. the powers of the quantity a ; which operation is called involution. Powers of the same quantity are multiplied by only adding their exponents, and making their sum the exponent of the product : thus, 'a4 x at = 09. Again, the rule for dividing powers of the same quantity is, to subtract the exponents, and make the difference the exponent of ad the quotient — = = • Negative powers, as well as positive, are multiplied by adding, and divided by subtracting, their exponents, as above. And, in general, any positive power of a, multiplied by a negative power of a, of an equal exponent, gives unit for the pro duct ; for the positive. and negative de stroy each other, and the product is to., which is equal to unit. Likewise, 5 a—.
= = a-3 a3; and — a-5 , 1 • And, in general, any quan.
tity placed in the denominator of a tion may be transposed to the numerator, if the sign of its exponent be changed : , 1 and 1 a-3, ano at.
a3 t."'"3 The quantity am expresses any power of a, in general ; the exponent on being undetermined: a—m expresses or a negative power of a, of an equal nent: and am 'X a—m = = a. = 1. Again, an expresses any other power am of a ; and a' X an = aten, and — an , To raise any simple quantity to its se cond, third, or fourth power, is to add its exponent twice, thrice, or four times, to itself; so that the second power of any quantity is bad by doubling its exponent ; and the third, by tripling its exponent ; and, in general, the power expressed by m, of any quantity, is had by multiplying the exponent by m: thus the second pow er, or square of a, is a' = a" ; its third power, = at ; and the mth power of a, is am ans. Also the square of a+, is a" 4 == ; the cube of a4, is and the mth power of a4, is The square of a b c, is a' b' a. ; its cube a3 63 co ; and the :nth power, ant bra cm.
PDX, or SMALL-Pox. See MEDICINE.