power or cube is a3× = a3; and the mth power of a is am×1 = a". Alfo, the fquare of a is a2x+= a; the cube of a is

the mth power of at is a4xm. The fquare of abc is a2bc, the cube is a b3c3, the mth power abc".

$41. The raifing of quantities to any power is called Involutions and any fimple quantity is. involved by multiplying the exponent by that of the power required, as in the preceding Examples.

The coefficient must also be raised to the fame power by continual multiplication of itelf by itself, as often as unit is contained in the exponent of the power required. Thus the cube of 3ab is 3 × 3 × 3 × a3b3 = 27 a3b3.

As to the Signs, When the quantity to be involved is pofitive, it is obvious that all its powers must be pofitive. And when the quantity to be involved is negative, yet all its powers whofe exponents are even numbers must be pofitive, for any number of multiplications of a negative, if the number is even, gives a pofitive; fince-xtherefore

= +,
= + ; and

+ x + x + = +.

- X

=33

+

The power then only can be negative when its exponent is an odd number, though the quantity to be involved be negative.. The powers of + a2, — a3, + a*, — a3,

-a are -a,

&c. Those whofe exponents are 2, 4, 6, &c. are pofitive; but thofe whofe exponents are 1, 3, 5, &c. are negative.

$42. The involution of compound quantities is a more difficult operation. The powers of any binomial a + b are found by continual multiplication of it by itfelf as follows.

a2 + 2ab + b2 = the Square or 2d Power.

+ ox

b

a* + 4a3b + + b

ba2b2 +

4ab3 + b+ =

Biquadrate or 4th Power.

a° + 5a3b + 10a+b2 + 10a3b3 +

+ ab + 5a4b2 + 10a3b3 + 10a2ba + 5abs + b6

a° + 6a3b + 15a*b* + 20a3b2 + 15a2b+ + 6ab3 + b 6th Power, &c.

5a2¿* + ab3

$43. If the powers of ab are required, they will be found the fame as the preceding, only the terms in which the exponent of b is an odd number will be found negative; "becaufe an odd number of multiplications of a negative produces a negative." Thus the cube of a-b will be found to be a 3ab2 b3 where the 2d and 4th negative, the exponent of b being an odd number in thefe terms. In general, In general," the terms of any power of ab are pofitive and negative by

turns."

3ab + terms are

$44. It is to be observed, that "in the first term of any power of ab, the quantity a has the exponent of the power required, that in the following terms, the exponent of a decreases gradually by the fame difference (viz. unit) and that in the laft term it is never found. The powers of b are in the contrary order; it is not found in the first term, but its exponent in the fecond term is unit, in the third term its exponent is 2; and thus its exponent increases, till in the laft term it becomes equal to the exponent of the power required."

As the exponents of a thus decrease, and at the fame time thofe of b increase, "the fum of their exponents is always the fame, and is equal to the exponent of the power required." Thus in the 6th power of a + b, viz. ao + бab + 15a+b2 + 20a3b3 + 15a2b* + 6abs + bo, D 3

the

the exponents of a 6, 5, 4, 3, 2, 1, 0;

decreafe in this order, and those of increase in the contrary order, 0, 1, 2, 3, 4, 5, 6. And the fum of their exponents in any term is always 6.

$45. To find the coefficient of any term, the coefficient of the preceding term being known; you are to "divide the coefficient of the preceding term by the exponent of b in the given term, and to multiply the quotient by the exponent of a in the fame term, increased by unit." Thus to find the coefficients of the terms of the 6th power of a + b, you find the

terms are

ao, a3b, aˆb3, a3b3, a2b1, ab3, bo;

and you know the coefficient of the first term is unit; therefore, according to the rule, the coefficient of the 2d term will be

× 5 + 1 = 6;

× 4+1=3×5

= 15; that of the 4th term will be

3

=5×4 = 20; and thofe of the following will be 15, 6, 1, agreeable to the preceding Table. $46. In general, if a + b is to be raised to any power m, the terms, without their coefficients, will be, a", a"—1b, a"—1b2, a”—3b3, a"—+b*, abs, &c. continued till the exponent of b becomes equal to m.

The

The coefficients of the refpective terms, according to the laft rule, will be

m-4, &c, continued until you have one co

5

efficient more than there are units in m.

It follows therefore by these last rules, that

m-2 m-3

3

4

× at −4 b4 + &c. which is the

general Theorem for railing a quantity confifting of two terms to any power m.

§ 47. If a quantity confifting of three, or more terms is to be involved, "you may dif tinguish it into two parts, confidering it as a binomial, and raise it to any power by the preceding rules; and then by the fame rules you may substitute inftead of the powers of thefe compound parts their values.”

-2

Thusa + b + c = a + b + c = a + b + 2 c x a + b + c2 = a2 + 2 a b + b2 + 2 ac+ 2 b c + c2.

And a + b + c3 = a + b3 + 3 6 × a + b2 + 3c2 xa + b + c3 = a2 + 3 a2 b + 3 ab2 + b2 3 b2 c + 3 a c2 + 3 b c2 + c2 •

× −

+ 3 a2 c + 6 a b c +