When unit is the greatest common measure of the numbers and quantities, then the fraction is already in its lowest terms. Thus 3ab

be reduced lower.

5dc

cannot

And numbers whofe greateft common meafure is unit, are faid to be prime to one another.

$34. If it is required to reduce a given fraction to a fraction equal to it that shall have a given denominator, you must multiply the numerator by the given denominator, and divide the product by the former denominator, the quotient fet over the given denominator is the fraction re

a

quired. Thus being given, and it being re

b

quired to reduce it to an equal fraction whose denominator fhall be c; find the quotient of ac divided by b, and it fhall be the numerator of the fraction required.

If a Vulgar fraction is to be reduced to a decimal (that is, a fraction whose denominator is 10, or any of its powers) annex as many cyphers as you please to the numerator, and then divide it by the denominator, the quotient fhall give a decimal equal to the Vulgar fraction propofed.

$35. Thefe fractions are added and fubtracted like whole numbers; only care mult be taken to fet fimilar places above one another, as units above units, and tenths above tenths, &c. They are multiplied and divided as integer numbers; only there must be as many decimal places in the product as in both the multiplicand and multiplier; and in the quotient as many as there are in the dividend more than in the divifor. And in divifion the quotient may be continued to any degree of exactnefs you please, by adding cyphers to the dividend. The ground of these operations is easily understood from the general rules for adding, multiplying, and dividing fractions.

CHAP.

CHAP. VII.

OF THE INVOLUTION OF QUANTITIES.

$36.

THE

THE products arifing from the continual multiplication of the fame quantity were called (in Chap. IV.) the powers of that quantity. Thus a, a, a3, aa, &c. are the powers of a; and ab, a2 b2, a3 b3, a* b*, &c. are the powers of a b. 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 axa' = a; and a3 b3 × a® b2 = a' b3. In Chap. V. 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."

§37. If you divide a leffer power by a greater, the exponent of the quotient must, by this Rule, be negative. Thus = a+ -" = a-2. But

2

a

and hence is expressed also by

6- with a negative exponent.

-3

a

= a°- = a;

20-3

aaa

I

I

4—3; so that the quantities a, 1,,,,

,&c. may be expressed thus, a1, ao, aTM', a—2, a−3, a−+, &c. -3 Thofe are called the negative powers of a which have negative exponents; but they are at the fame time pofitive powers of

a

or a1.

§ 38. Negative powers (as well as pofitive) are multiplied by adding, and divided by fubtracting their exponents. Thus the product of aTM2 (or

multiplied by a (or) is a—4—3—as

(or —;) ;' also a—ˆ × a* = a−6++ = a−2 (or

——); and a~3 × a1 = a° = 1. And, in gene= ral, 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 deftroy each other, and the product gives a°, which is equal to unit.

Likewise == a−s +2 = 67.2 =

ral, "

I

and

; therefore a': And, in gene

-3

=

any quantity placed in the denominator of a fraction may be tranfpofed to the numerator, if the fign of its exponent be changed."

$39. The quantity a" expreffes any power of a in general; the exponent (m) being undetermined; and a-" expreffes or a negative

I

power of a of an equal exponent: and a" × a= a"" = a° = 1 is their product. a" expreffes any other power of a; a" x a"a"+" is the product of the powers a" and a", and a"-" is their quotient.

$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 of any quantity is bad by multiplying the exponent by m, as is obvious from the multiplication of powers. power or fquare of a is ax

Thus the fecond = a'; its third

power