II. To prove that

am

an

am-n universally, m and n being in

tegers.

let

1. Suppose the exponent of the numerator to be negative; or

2. Suppose the exponent of the denominator to be negative; or

mn

III. To prove that (am)" = am" universally, m and n being in

3. Suppose both m and n to be negative; or let

m = -m' and n = —

-n'

- n'.

n'

Then (a")" = (a ̄")—"' = ( ~—-) ̃" = (~7′′)"=

= am'n' = a'

mn

Hence, in all algebraic operations, the same rules will apply to negative exponents as to positive. That is, if two powers of the same quantity be given, then the exponent of their product will be equal to the algebraic sum of the given exponents, and the exponent of their quotient will be equal to the algebraic difference of the given exponents.

EXAMPLES.

215. Find the value of each of the following expressions:

POWERS OF POLYNOMIALS.

216. A polynomial may be raised to any power by actual multiplication. Thus, if the quantity be multiplied by itself, the product will be the second power; if the second power be multiplied by the quantity, the product will be the third power; and Hence the following

so on.

RULE.-Multiply the quantity by itself in continued multiplication, till it has been taken as many times as a factor as there are units in the exponent of the required power.

NOTE.-It may be well to observe that in involution we may often reach the same result by different processes. Thus, we have a=a3 × a=a1× a2 = (a3)2=(a2)3.

1=

4. (3a+2b+c)3.

Ans. 27a+54a2b+27a2c+36ab2+36abc+863+9ac2+1272c+. Sbc2 + c3.

5. (a + b)".

Ans. a + ab + 21a5b2 + 35a4b3 + 35a3b1+ 21a2b5+ 7ab®+b%.

Ans. x3- 8x3y + 28x©y2 — 56x3y3 + 70x1y1 — 56x3y3 + 28x2y® — 8xy' + y3.

7. (a2c ̄2 + a ̄2c2)2.

Ans. ac + 2 + a ̄1c2.

8. (a2+1+a-2)3. Ans. ab+3a2+6a2+7+6a2+3a ̄1+a ̄€. 9. (am + xn)3.

Ans. asm+3a2x2 + 3amx2n + x3n ̧

POLYNOMIAL SQUARES.

217. We have seen that the square of any binomial may be written without the labor of formal multiplication (70). Thus, if x and y represent the terms of any binomial, then

(x + y)2 = x2 + 2xy + y2.

This formula for a binomial square furnishes a simple rule for writing the square of any polynomial, in the same direct manner. To deduce the method, let it be required to square the polynomial,

a+b+c+d+e+....

Put xa and y = b+c+d+e+

....

Then the

x+y will be equal to the square of the given polynomial; or

x2+-2xy + y2=(a+b+c+d+e+ ....)2.

And the three parts of the required square will be

Now y represents a polynomial; and to obtain its square, we must proceed as at first. Thus, put 'b and y' =c+d+e+

...

Then the square of x'+y' will be equal to the square of b+c+ d+e+.... And we have

1. Square a+b+c. Ans. a2+2ab + 2ac + b2 + 2bc + c2. 2. Find the square of a + b + c + d.

Ans. a2+2ab+2ac+2ad+b2+2bc+2bd+c2+2cd+d2.

3. Find the square of a+b+c+d+e.

Ans. a2+2ab+ 2ac + 2ad + 2ae + b2 + 2bc + 2bd + 2be + c2+2cd+ 2ce + d2 + 2de + e2.

Ans. a2-4ab+6a2b-2ac+4b2-12ab2+4bc9a2b2-6abc+c2.

8. Find the square of 1-2x - y2 + xy — x2.

· 2y2 + 2xy + 2x2 + 4xy2 - 4x2y + 4x3 + y$ —

218. In a future section we shall give a formula, called the Binomial Formula, by means of which any power of a binomial may be obtained without the labor of multiplication.

Th

pr

Ans. 27a3+ 54a2b +VOLUTION.

6bc2+ c3.

5. (a+b)

Ans. a

of any quantity is one of the equal factors together, will produce the given quantity.

name or degree of a root corresponds to the num6. ( factors into which the quantity is supposed to be

Ans.

8xy

al

Thus,

square root of a is one of the two equal factors whose duct is a.

The cube root of a is one of the three equal factors whose product is a.

The fourth root of a is one of the four equal factors whose product is a; and so on.

221. Evolution is the process of extracting any root of a given quantity; it is the converse of involution.

222. There are two methods of indicating evolution :

1st. By the radical sign, √.

When this method is employed, the name or degree of the root is denoted by a figure or letter written above the radical, called the index of the root. Thus, Va denotes the cube root of a; and a denotes the fourth root of a. When no index is written, 2 is understood. Thus, V denotes the square root of , and signifies the same as V.

2d. By fractional exponents.

To explain the origin of this method of indicating roots, we observe that a quantity is raised to any power, by multiplying its exponent by the exponent of the required power. Conversely, any root of a quantity may be obtained, by dividing the exponent of the quantity by the index of the required root. Thus, the cube root of a, or a1, is written a3, and the cube root of a2 will be a3.

Hence, a fractional exponent may be analyzed as follows:

1. The numerator denotes the power of the quantity, whose root is to be extracted.

2. The denominator shows what root of that power is to be extracted.