CHAP. VII.

Of the INVOLUTION of Quantities.

$ 36.THE

$36.HE products arifing from the continual multiplication of the fame quantity were called (in Chap. 4.) the powers of that quantity. Thus a, a, a3, a*, &c. are the powers of a; and ab, a2b2, a3b3, a+b+, &c. are the powers of ab. In the fame Chapter, the rule for the multiplication of powers of the fame quantity is to "Add the exponents and make their fum the exponent of the product." Thus at X a = a; and a3b3 × a°b2 = ab3. In Chap. 5. you have the rule for dividing powers of the fame quantity, which is, " To fubtract the exponents and make the difference the exponent of the quotient."

Thus as

= a2 ; and

a563

a+b

as-4b3—— ab2.

$37. If you divide a lesser power by a greater, the exponent of the quotient must, by this Rule,

be negative.

26

;

a4

Thus = a4—6 = a-2. But 26

and hence is expressed also by a2

with a negative exponent.

α

aaa

a-3; fo that the quantities a, I, a' az9 a33

2

938

&c. may be expreffed thus, a', ao, a—ı

a-2, a-3, a-4, &c. Thofe are called the negative powers of a which have negative exponents; but they are at the fame time pofitive powers of

a

or a-1.

§ 38. Negative powers (as well as pofitive) are multiplied by adding, and divided by fubtracting their exponents. Thus the product of a-2 (or

~) multiplied by a―3 (or 1) is a—2—3—a—

=a-s

(or 1); also a− × a* = a−6+4 = a−2 (or

3

1); and a—3 × a3 = a° = 1. And, in general, any pofitive power of a multiplied by a negative power of a of an equal exponent gives UNIT for the product; for the pofitive and negative destroy each other, and the product gives a°, which is equal to unit.

=

= a−2+5 = a3. But alfo,=

I

= 1; therefore == a3: And, in general,

a-3

a-3

any quantity placed in the denominator of a fraction may be transposed to the numerator, if the fign of its exponent be changed." Thus a and ——, = a3.

-3

a

I

3

$ 39. The quantity a expreffes any power of a in general; the exponent (m) being unde

I

termined; and a-m expreffes or a negative

am

power of a of an equal exponent: and am xa-m = am—m = a° = 1 is their product. a" expreffes

any other power
of a;
am Xan =
product of the powers am and a",
their quotient.

am+n is the

and am—" is

§ 40. To raise any fimple quantity to its fecond, third, or fourth power, is to add its exponent twice, thrice, or four times to itfelf; therefore the fecond power of any quantity is had by doubling its exponent, and the third by trebling its exponent; and, in general, the power expreffed by m if any quantity is had by multiplying the exponent by m, as is obvious from the multiplication of powers. Thus the fecond power or fquare of a is a2x1 = a2; its third

power