Thus, 4 Or, X a = =a2, Or, _-) - 1 (~ —a2. I. Of Involution. To find any power of any quantity, is the business of involution. Cafe I. When the quantity is fimple. Rule. Multiply the exponents of the letters by the index of the power required, and raife the coefficient to the fame power. Thus, the 2d power of a is a The 3d power of 2a2 is 8a 2X3 =a2. =8a. 6x3 The 3d power of 3ab' is 27a1×3b3×3—27a3b3. For the multiplication would be performed by the continual addition of the exponents; and this multiplication of them is equivalent. The fame rule holds alfo when the figns of the exponents are negative. 1 If the fign of the given quantity is +, all T its powers must be pofitive. If the fign is then all its powers whofe exponents are even numbers, are pofitive; and all its powers whose exponents are odd numbers, are negative. This is obvious from the rule for the figns in multiplication. The laft part of it implies the most extenfive use of the figns + and -, by fuppofing that a negative quantity may exift by itself. Cafe II. When the quantity is compound. Rule. The powers must be found by a continual multiplication of it by itself. multiplying the fquare already found by the root, &c. -4 as atitwoup moving sds to 2 Fractions are raised to any power, by raifing both numerator and denominator to that power, as is evident from the rule for multiplying fractions, in chap. 1. p. 2. The involution of compound quantities is rendered much easier by the binomial theorem; for which, fee Chap. VII. Sect.3. Note. The fquare of a binomial confifts of the fquares of the two parts, and twice the product of the two parts. II. Of Evolution. Evolution is the reverfe of involution, and by it powers are refolved into their roots. Def. The root of any quantity is expreffed by placing before it (called a radical fign) with a fmall figure above it, denoting the denomination of that root. Thus, 2 a Thus, the fquare root of a is or , The cube root of be is /bc. The 4th root of a2bx3 is £q2b÷x3• yilli 5. The mth root of cdx is d elii, qui ne baileɔ si ti audt bas ytherop General Rule for the Signs. ༧༽སཙྩུ 1. The root of any pofitive power may be cither pofitive or negative, if it is denomi nated by an even number; if the root is denominated by an odd number, it is pofitive only. 2. If the power is negative, the root alfo is negative, when it is denominated by an odd number. 3. If the power is negative, and the dengmination of the root even, then no root can be affigned. This rule is eafily deduced from that given in involution, and fuppofes the fame extensive use of the figns + and - If it is applied to abstract quantities in which a contrariety cannot be fuppofed, any root of a pofitive quantity must be pofitive only, and and any root of a negative quantity, like it. self, is unintelligible. 210 to 1001 sdub silT In the laft cafe, though no root can be affigned, yet sometimes it is convenient to fet the radical figns before the negative quantity, and then it is called an impossible or imaginary root, ut au lorons) The root of a pofitive power, denominated by an even number, has often the fign± before it, denoting that it may have either + or The radical fign may be employed to exprefs any root of any quantity whatever; but fometimes the root may be accurately found by the following rules, and when it cannot, it may often be more conveniently expreffed by the methods now to be explained. Cafe I. When the quantity is fimple. Rule. Divide the exponents of the letters by the index of the root required, and prefix the root of the numeral coefficient.salgga 1. The exponents of the letters may be multiples of the index of the root, and the |