Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/117

 85 A L G I ^ B R A. '—ax ; its third power or cube is a 3X'^ra?; and the4 Thus, p-=ra4~.<5 = « —*. But, ~ ^ ; and a"«th^power of a is a '"X =a”'. Alfo, the fquare of a hence ~ is exprefled alfo by a1 with a negative ex- is Vaa IX4=ra8 • the4 cube of a4 is a3 X4=:a,x ; and the m1 xpower The fquare of a £ c is 1 of a is a 3 W. ponent. a b c, the cube is a b3 cs, the zwth power ambm cvt. It is alfo obvious, that —a — a' ~xzzaa ; but—a = x, The raifing of quantities to any power is called invoand any fimple quantity is involved by multiplyand therefore a0 = x. After the fame manner —a = —a = lution, ing the exponent by that of tire power required, as in the preceding examples. The coefficient mufl alfo be raifed to the fame power by a continual multiplication of itfelf by itlelf, as often — <*°—} =: a—3; fo that the quantities a, i, —, —’ as unit is contained in the exponent of the power3requi3 1 -1 red. 3 Thus the cube of 3 a £ is 3 x 3 x 3 x a £ = aJ-,* 1 a—,4 3 fee. 4may be expreffed thus, a, a°, a , 27Asa tob*. the figns. When the quantity to be involved is a-", a— , a— , frc. Thofe are called the negative it is obvious that all its powers mufi be poftive. powers of a which have negative exponents ; but they poftive, And, when the quantity to be involved is negative, yet all its powers, whofe exponents are even numbers, muf be poare at the fame time pofitive powers of — or a~x. ftive
 * for any number of multiplications of a negative,

Negative powers (as well as pojitive) are multiplied if the number be even, gives a pofitive; fince — x — by adding, and divided 1by fubtrafling their exponents. = +, therefore — x-^—x — x — = + x + = +; and — x — x — x — x — x — = + x + x + = +. Thus the product of a— (or multiplied by The power then only can be negative when its expo-1 3 s 6 nent odd number, though the quantity to be invol-1 (or is tf —' =: a — (or -i- ; ^ alfo a— X ved 3beis annegative. The powers of — a are -— a, + a, — a , +«4, —a5, ere. Thofe whofe exponents are 3 a*3 = a — 6 + 4= «“"* (or-L;j and a — X 2, 4, 6, isc. are pofitive; but thofe whofe exponents 5, 6v. are negative. a — a° — i. And, in general, any pojitive power of are 1, 3,involution of compound quantities is a more diffia multiplied by a negative power of of an equal expo- cultTheoperation. The powers of any binomial a + 3 are nent gives unit for the produft; for the pofitive anda found by a continual multiplication of it by itfelf, as fol• negative deftroy each other, and the produdt gives a , lows : which is equal to unit. 3 a + 3 = Root. Likewife a—* -—•, =r a— c + 2 = a~ = a3 ; and a— —-3 = Xa + 3 5 =—1 a — 2+y = a3. But alfo, ^—? a1 + a 3 a— a— X a—3 = + a3 + i* } 1 —therefore
 * And, in general, “ A1

a J a-^—.zra « + 2«3 + 3xcr the fquare or 2d power* “ ny quantity placed in the denominator of a fraction “ may be tranfpofed to the numerator, if the3 fign of its X*3 +3 x a + 21a 3 + a 31l 3 “ exponent be changed.” Thus = «~, and —3 +a 3+2a3 +3 3 =r a. 3 The quantity am exprelfes any power of a in general, a + 3 as 3 + 3 a 3X + 33 = cube or 3d power, <bc. m the exponent (m) being undetermined; and ar— exIf the powers of a — 3 are required, they will be prefies —, orma negative power of a of an equal expo- found the fame as the preceding, only the terms in which ponent: nand a x a—m — am—m~a° ~ i is their pro- the exponent of 3 is an odd number, will be found nedud:. m n a exprefies any other power of ma ; a’" X a" = gative ; “ becaufe an odd number of multiplications of a negative a -- ;s the projdud of the powers a and a”, and a“ — 3 will beproduces found toabenegative.” a3 — 3 a* Thus, 3+3 athe3Z cube — 33of: am—n js theij- quotient. the 2d and 4th terms are negative, the exponent To raife any fimple quantity to its fecond, third, or Where 3 being an odd number in thefe terms. In general, fourth power, is to add its exponent twice, thrice, or of“ The terms of any power of a — 3 are pofitive and four times to itfelf; therefore the fecond power of any “ negative turns.” quantity is had by doubling its exponent, and the third It is to bebyobferyed. That “ in the firft term of any by trebling its exponent; and, in general, the power power of a 3, the quantity the exponent of exprejfed by m of any quantity is had by multiplying the the power required; that in thea has' terms, the exponent by m, as is obvi us from the multiplicat on of exponents of a decreafe gradually following by the fame differpowers. Thus the fecond power or fquare of a is Y ence Von. I, No. 4. 3