THEOREM XIV.

of

170. If any quantity, 2a, be divided into two equal quantities, a and a; and, if any quantity, b, be added to 2a, the square 2a+b, plus the square of b, will be equal to twice the square of a, plus twice the square of a+b.

Now

But

Therefore

(2a+b)2+b2=4a2+4ab+b2+b2=4a2+4ab+2b2.
4a2+4ab+26=2a2+2(a+b)2.
(2a+b)2+b2=2a2+2(a+b)2.

ART. 171. Involution is the raising of powers from any proposed root; or, the method of finding the square, cube, biquadrate, &c., of any given quantity.

172. A power is the product of any quantity multiplied into itself a certain number of times, and the degree of the power is denoted by an exponent written over the root. Thus as is the third power of a, and a is the root.

173. The exponent, or index, shows how many times the root has been used as a factor.

Thus,

axaxaxa=a', and xXx=x2.

174. When a quantity is written without any index, its index is uniformly considered a unit. Thus, a—a1, and x=x1. There

fore, to raise any quantity to any required power, the pupil will see the propriety of the following

RULE. 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 power required.

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

175. When the sign of any simple quantity 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.

EXAMPLES.

1. What is the fifth power of a?

2. What is the third power of ax?

3. Required the square of a2x.

4. Required the cube of +3a2.

5. Required the fourth power of —ab2c3.

6. Required the square of —

2ax2
3b.*

7. Required the fifth power of 2ab2x3. 8. Required the sixth power of fa3x% 9. Required the third power of 2a. 10. Required the fourth power of -3m-3. 11. Required the mth power of a”. 12. Required the fourth power of 2xTM.

176. Polynomials are involved by multiplying the quantity

by itself as many times, wanting one, as there are units in the exponent of the power.

3d power.

(a+b)3=a3+3a2b+3ab2+b3
a+b

a1+3a3b+3a2b2+ab3
+ab+3a2b2+3ab3+b*

(a+b)1=a1+4a3b+6a2b2+4ab3+ba
a+b

a5+4a1b+6a3b2+4a2b3+ab1

+ab+4a3b2+6a2b3+4ab1+b3

(a+b)=a+5a1b+10a3b2+10a2b3+5ab1+b5

Required the third power of a-b.

(a—b)'=a-b
a -b

a2-ab

Ans. x3—10x1y+40x3y2—80x3y3+80xy^—32y3.

16. Required the third power of a−b+1.

Ans. a3-3a2b+3a2+3ab2-6ab+3a-b3+362-36+1.

17. Required the second power of 2x2—3x+4.

Ans. 4x-12x+25x2-24x+16.

18. Required the sixth power of x-2.

Ans. x-12x+60x1—160x3+240x2-192x+64.

20. Required the fourth power of a”—a”.

Ans. am—4a3m+n+6a2m+2n—4ɑm+3n+a1.

21. What is the second power of 2x2-3x+1?

Ans. 4x-12x+11x2-3x+1.

22. What is the third power of a+2b—c?

Ans. a3+6a2b-3a2c+12ab2-12abc+3ac2+863—1262c+6bc3

23. What is the fourth power of a+b+c+d?

Ans. a1+4a3b+6a2b2+4ab3+b2+4a3c+12a2bc+12ab2c+463c

+4a3d+12a3bd+12ab3d+4b3d+6a2c2+12abc2 + 662c2 + 12a2cd +24abcd+1262cd+6a2d2+12abd2 + 6b3ď2+4ac3 +12ac3d + 12acd2+4ad3+4bc3+12bc3d+12bcd2+4bd3+c*+4c3d + 6c3ď2+

4cd3+d".

24. What is the second power of x3+2x2+x+2? Ans. x+4x+6x1+8x3+9x2+4x+4.

26. What is the third power of x2—x—1?

Ans. x-3x+5x3-3x-1.

27. What is the third power of a—b—2c2—d3 ?

Ans. a3—3a2b+3ab2—b3—6a2c2+12abc2—6b3c2—3a2d3+6abd3 -36'd3+12ac'+12ac d3+3ad-12bc-12bc'd3-3bd — 8c — 12c+d2-6c2d-do.

SECTION XIV.

EVOLUTION, OR THE EXTRACTION OF ROOTS.

ART. 177. Evolution is the reverse of involution, being the method of finding the roots of any given quantity. It will, therefore, be necessary to trace back the steps of the operation in involution.

Hence, to find any root of a monomial, we adopt the following

RULE. Extract the required root of the coefficient for the coefficient of the answer, and the root of the quantity subjoined for the literal part of the answer.

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

179. The square root, the fourth root, or any other even root of an affirmative quantity, may be either plus or minus.

Thus, a2=+a or -a; and ba=+b, or —b. But the cube root, or any other odd root of a quantity, will have the same sign as the quantity itself. Thus

aa;

a3——a,

The reason why +a and ―a are each the square root of a2, is obvious; since, by the rule of multiplication, (+a)X(+a) and (—a)X(—a) are each equal to a2.

180. In the case of the cube root, fifth root, &c., of a negative quantity, the rule is equally plain; since, by multiplying, we have (-a)X(-a)X(-a)——a3.

It may also be stated here that any even root of a negative quantity is unassignable; or, as it is usually called, imaginary. Thus, a cannot be determined, as there is no quantity, either positive or negative, that, when multiplied by itself, will produce -a2.