THE PRODUCT OF THE SUM AND DIFFERENCE OF TWO QUANTITIES, IS EQUAL TO THE DIFFERENCE OF THEIR SQUARES.

This is another instance of the facility with which general truths are demonstrated in algebra. See arts. 23 and

77.

If the sum and difference of the squares be multiplied, the product will be equal to the difference of the fourth powers, &c.

Thus (ay)x(a+y)=a3—y3.

(a2 —y3)X(a2+y2)=a*—y'.
(a1-y1) x (a+y1)=a® - y®, &c.

DIVISION OF POWERS.

236. Powers may be divided, like other quantities, by rejecting from the dividend a factor equal to the divisor; or by placing the divisor under the dividend, in the form of a frac

tion.

Thus the quotient of a3b2 divided by b2, is a3. (Art. 116.)

Divide 9a3y4

α

dx(a-h+y)3 (a-h+y)?

if any term be divided by another, the index of the quotient will be equal to the difference between the index of the dividend, and that of the divisor.

237. A POWER MAY BE DIVIDED BY ANOTHER POWER OF

THE SAME ROOT, BY SUBTRACTING THE INDEX OF THE DIVISOR FROM THAT OF THE DIVIDEND.

yyy

Thus y3÷y3=y3 ̄2=y'. That is YYY=y.

238. The rule is equally applicable to powers whose ex

ponents are negative.

The quotient of a ̄5 by a ̃3, is a ̄3.

1

1

3. hahhati-h3. That is ha

That is

=

h
·=h2×==h:

h

4. 6a2a3=3a"+3.

5. ba3a bao.

8. (a3+y3)m÷(a3+y3)"=(a3+y3)m ̄". 9. (b+x)”÷(b+x)=(b+x)” ̄1,

The multiplication and division of powers, by adding and subtracting their indices, should be made very familiar; as they have numerous and important applications, in the higher branches of algebra.

EXAMPLES OF FRACTIONS CONTAINING POWERS.

239. In the section on fractions, the following examples were omitted for the sake of avoiding an anticipation of the subject of powers.

4. Reduce 8a3y-12a3y2+6ay to lower terms.

Ans.

3

4a3 −6ay+3y2 obtained by dividing each term by 2ay.

3a+2y

a2 Xa 4 is a 2, the first numerator. (Art. 146.)
ας χα
аз × а 3 is α=1, the second numerator.

SECTION IX.

EVOLUTION AND RADICAL QUANTITIES.*

ART. 240. IF a quantity is multiplied into itself, the product is a power. On the contrary, if a quantity is resolved into any number of equal factors, each of these is a root of that quantity.

Thus bis a root of bbb; because bbb may be resolved into the three equal factors, b, and b, and b.

In subtraction, a quantity is resolved into two parts.

In division, a quantity is resolved into two factors.

In evolution, a quantity is resolved into equal factors.

241. A ROOT OF A QUANTITY, THEN, IS A FACTOR, WHICH MULTIPLIED INTO ITSELF A CERTAIN NUMBER OF TIMES, WILL PRODUCE THAT QUANTITY.

The number of times the root must be taken as a factor, to produce the given quantity, is denoted by the name of the

root.

Thus 2 is the 4th root of 16; because 2×2×2×2=16, where 2 is taken four times as a factor, to produce 16.

So a3 is the square root of a°; for a3 Xaa=a®. (Art. 233.) And a2 is the cube root of a6; for a2 Xa2 Xa2=ao. And a is the 6th root of a; for a XaXaXa×a×a=a®. Powers and roots are correlative terms. If one quantity is a power of another, the latter is a root of the former. As b3 is the cube of b; b is the cube root of b3.

242. There are two methods in use, for expressing the roots of quantities; one by means of the radical sign✔, and the other by a fractional index. The latter is generally to be preferred; but the former has its uses on particular occa

sions.

*Newton's Arithmetic, Maclaurin, Emerson, Euler, Saunderson and Simpson.

When a root is expressed by the radical sign, the sign is placed over the given quantity, in this manner, ✔a.

Thus

a is the 2d or square root of a.

3a is the 3d or cube root.

"Va is the nth root.

And "ay is the nth root of a+y.

243. The figure placed over the radical sign, denotes the number of factors into which the given quantity is resolved; in other words, the number of times the root must be taken as a factor, to produce the given quantity.

The figure for the square root is commonly omitted; ✔a being put for a. Whenever, therefore, the radical sign is used without a figure, the square root is to be understood.

244. When a figure or letter is prefixed to the radical sign, without any character between them; the two quantities are to be considered as multiplied together.

Thus 2a, is 2x a, that is, 2 multiplied into the root of a, or, which is the same thing, twice the root of a.

And xb, is x xb, or x times the root of b.

When no co-efficient is prefixed to the radical sign, 1 is always to be understood; a being the same as 1a, that is, once the root of a.

245. The method of expressing roots by radical signs, has no very apparent connection with the other parts of the scheme of algebraic notation. But the plan of indicating them by fractional indices, is derived directly from the mode of expressing powers by integral indices. To explain this, let a be a given quantity. If the index be divided into any number of equal parts, each of these will be the index of a

root of a.

6

Thus the square root of a is a3. For, according to the definition, (Art. 241,) the square root of a is a factor, which multiplied into itself will produce a°. But a3 Xa3=a*. (Art. 233.) Therefore, a3 is the square root of a. The

index of the given quantity a°, is here divided into the two equal parts, 3 and 3. Of course, the quantity itself is resolved into the two equal factors, a3 and a3.