RULE I.

Multiply the quantity into itself as many times as is denoted by the index of the power less one, and the last product will be the power required.

Or, multiply the index of the quantity by the index of the power to which it is to be raised, and the result will be the same as before.

Note. When the sign of the root is +, all the powers of it will be + ; and when the sign is —, all the even powers will be +, and the odd powers - as is evident from multiplication (t).

square and cube; but when they are of a higher kind, they are usually mentioned by the terms fourth power, fifth power, and so on, according to the index by which they are denoted.

(t) Any power of the product of two or more quantities is equal to the same power of each of the factors multiplied together: thus,

(ub)m = am × bm; and (abc)maTM × bTM × cm.

And any power of a fraction is equal to the same power of the numerator divided by the like power of the denominator: thus,

Also, a raised to the nth power is am2; and a raised to the ath power is a", according as n is an even or an odd number.

x3 — 3 ax2 + 3a2x — a3 cube. | x2+3ax2+3a2x+a3 cube.

EXAMPLES FOR PRACTICE.

1. Required the cube, or third power, of 2a2. 2. Required the biquadrate, or 4th power, of 2a x. 3. Required the cube, or third power, of - 12x'y3.

4. Required the biquadrate, or 4th power,

of

3a2x

562

5. Required the 5th power of a + x; and the 6th power of a-y.

RULE II.

A binomial or residual quantity, may also be readily raised to any power whatever, as follows:

1. Find the terms without the coefficients,, by observing that the index of the first, or leading quantity, begins with that of the given power, and decreases continually by 1, in every term to the last; and that in the following quantity, the indices of the terms are 1, 2, 3, 4, &c.

2. To find the coefficients, observe that those of the first and last terms are always 1; and that the coefficient of the second term is the index of the power: and for the rest, if the coefficient of any term be multiplied by the index of the leading quantity in it, and the product be divided by the number of terms to that place, it will give the coefficient of the term next following.

,

Note. The whole number of terms will be one more than the index of the given power; and when both terms of the root are + all the terms of the power will be +; but if the second term be all the odd terins will be +, and the even terms; or, which is the same thing, the terms will be and alternately (u).

(u) The rule here given, which is the same, in the case of integral powers, as the binomial theorem of Newton, may be expressed in general terms, as follows:

EXAMPLES.

1. Let x+a be involved, or raised to the 5th

power.

Here the terms, without the coefficients, are

x3, xa, x'a, x'a', xa*, a'.

And the coefficients will be

5,

1.

or 1, 5, 10,

10,

Whence the entire 5th power of a + x is

ta,

x+5xa+10x a2+10x a3 + 5xα1 + α3,

or x2+5ax*+ 10a2x2 + 10a3x2 + 5α*x + a3. 2. Let a-x be involved, or raised, to the 6th power.

Here the terms, without their coefficients, are

ao, a3x, a*x2, a3x3, a°x1, ax3, xo.

And the coefficients are

Which formule will, also, equally hold when m is a fraction, as will be more fully explained hereafter.

It may, also, be farther observed, that the sum of the coefficients in every power, is equal to the number 2 raised to that power. Thus, 1+1=2, for the first power; 1+2 + 1 = 4 = 22, for the square; 1 + 3 + 3 + 1 = 8 = 23, for the cube, or third power; and so on.

The trinomial quantity a+b+c may also be raised to any power in the same way, by considering two of its terms as a single one, and proceeding as before. Thus,

(a + b + c)3 = a3 + 3(a + b)a2 + 3 (u + b)2a + (a + b)3 = a3 + b3 + c3 — 3abc + 3(ab + ac + bc) × (a + b + c).

(a + b + c)TM = aTM + m(b + c)aTM~1 + m.

m

3

2

m

-

1

And

m 1

(b + c)m −2a + m.

2

2

(b + c)3am-3, &c.; where the powers of b + e may be deter

mined by the general theorem, as usual.

Whence the entire 6th power of a-x is

ao - 6a3x + 15a2x2 - 20a3x3 + 15a2x2 - 6ax3 + x 3. Required the 4th power of a+x, and the 5th power of a-x.

4. Required the 6th power of b+x, and the 7th power of b-y.

5. Required the 5th power of 2+x, and the cube of a- bx + c.

EVOLUTION.

(H) EVOLUTION, or the extraction of roots, is the reverse of involution, or the raising of powers; being the method of finding the square root, cube root, &c. of any given quantity.

CASE I.

To find the roots of any powers of a simple quantity.

RULE.

Extract the root of the coefficient for the numeral part, and the root of the letter, or letters, for the literal part; and these, joined together, will be the root required.

And if the quantity proposed be a fraction, its root will be found, by taking the root both of its numerator and denominator.

Note. The square root, the fourth root, or any other even root, of an affirmative quantity, may be either + or Thus,

√ a2 = + a ora, and b+b or -b, &c.