Page images


Thus, 4-8-201

Or, x=

=a*, Or, 10=a.

[ocr errors][merged small]

To find any power of any quantity, is the business of involution,

Cafe I. When the quantity is simple.

[ocr errors]


[ocr errors]

Rule. Multiply the exponènts of the letters

by the index of the power required, and

raise the coefficient to the same power.
Thus, the 2d power of a is a
The 3d power of 2a is 8a
The 3d power of zab?is 27a******

For the multiplication would be perform-
ed by the continual addition of the expo-
nents; and this multiplication of them is
equivalent. The same rule holds also when
the signs of the exponents are negative.


y .!" I rict Rules for the Signs. 3 VAR? If the sign of the given quantity is t, all

must be positive. If the sign is then all its powers whose exponents are even numbers, are positive; and all

powers whose exponents are odd numbers, are negative.

its powers muijt be politine te


This is obvious from the rule for the signs in multiplication.

The last part of it implies the most extensive use of the signs + and — by fuppofing that a negative quantity may exist

by itfelf.

Case II. When the quantity is compound.

Rule. The powers must be found by a conti... nual multiplication of it by itself.

Thus, the square of x+is found by multiplying it into itself. The product is




multiplying the square already found by the root, &c. mitsus

V Fractions are raised to any power, by rai2 sing both numerator and denominator to that power, as is

as is evident from the rule for multiplying fractions, in chap. I. p. 2.

The involution of compound quantities is rendered much easier by the binomial theorem ; for which, see Chap. VII. Sect.3.

Note. The square of a binomial consists of the squares of the two parts, and twice the product of the two parts.

II. Of Evolution.

Evolution is the reverse of involution, and by it powers are resolved into their


Def. The root of any quantity is expressed by placing before it ✓ (called a radical fign) with a small figure above it, denoting the denomination of that root.


[ocr errors]

Thus, the square root of n is ā or you

The cube root of b¢ is bc. .: JUOTI

3. Aladin o The 4th root of ab**** js Z4?6**igi.. 9. The mth root of cuadx is idxb

a boiliei si ft boe is

Gis :

General Rule for the Signs. 1. The root of any positive power may be eia

ther positive or negative, if it is denominated by an even number; if the root is denominated by an odd number, it is pofi

tive 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 deno

mination of the root even, then no root can be assigned.

This rule is easily deduced from that given in involution, and suppofes the same extensive use of the signs + and If it is applied to abstract quantities in which a contrariety cannot be supposed, any root of a positive quantity, must be positive only,


and any root of a negative quantity, like it self, is unintelligible.

2191 10 1001 9:10T In the last case, though no root can be assignéd, yet sometimes it is convenient to set the radical signs before, the negative quantity, and then it is called an imposible or imaginary root, so borili?

The root of a positive power, denominated by an even number, has often the sign before it, denoting that it may have either + or

The radical sign may be employed to express 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 expressed by the methods now to be exa plained.

[ocr errors]

Cafe I. When the quantity is simple. Rule. Divide the exponents of the letters by :ibe index of the root required, and prefix the foot of the numeral coefficient. ,

1. The exponents of the letters may be' multiples of the 'index of the root, and



« PreviousContinue »