To find any power

of

any quantity, is the business of involution.

Cafe I. When the quantity is simple.

ix2

2X3

Rule. Multiply the exponents 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 zaż is 8a%**=8a.
The 3d power of zab’ is 27a****=27ab.

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.
L

Rule

82
* ! ! Rule' for the Signs. 17 ***

Dit
If the sign of the given quantity is t, all

Di hosil
its powers must be positive. If the sign is

then all its powers whose exponents are even numbers, are positive ; and all its powers whose exponents are odd numbers, are negative.

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

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

by itself.

Case II. When the quantity is compound.

Rule. The powers must be found by a conti

nual multiplication of it by itself. ;

a

Thus, the square of x+ is found by

xt multiplying it into itself. The product is m2 + axta*. The cube of xta is got by gja

2 od multiplying

а

X

4

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

timilinu.91g as

Fractions are raised to any power

so any power, by rai

sing both numerator and denominator to

10000 Puso 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 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 roots.

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.

Thus,

2

Thus, the square root of a is vā or you

The cube root of be is bc. to Join Miller

sos adial The 4th root of a brex? is 4?b=**.211).

* 9: The mth root of candx is it Sitios ::

res boilez et ji (10:{t bor

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

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 car be asigned.

This rule is easily deduced from that given in involution, and supposes 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

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

10 1001 sdu In the last case, though no root can be affignéd, yet Tometimes it is convenient to set the radical signs before, the negative quantity, and then it is called an impossible or imaginary root, sorgen

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 sometimes 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 explained.

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

**The exponents of the letters may be' multiples of the index of the root

the

I.

root, and