(2x-3y)3 = 8x3 — 36x2y +54xу2 — 27y3

15. Find the cube of + sy.

(2ax — 3by2)3 = 8a3x3 — 36a2bx2y2 + 54ab2xy1 =8a3x3-36a2bx2y2 — 27b3y.

18. Find the fourth

power of

ax су

d

19. Find the fourth power of mx + y.

Ans. m11+4m3nx3y + 6m2n2x2y2+ 4mn3xy3 † n1y*

=e3x

3x

32αx3+16

1.

24. (e-e-x)3 = e3z − e−3* — 3(e* — e-x).

.

25. (5-4)+ = 625-2000x + 2400x2-1280x3+ 256x

106. The polynomial a+b+c, may be written under

the form, a+b+c); hence, (a+b+c)"

also, (2α-x+3y+4z)", equal to [(2a-x)+(3y +4z)}* ;

and so on, for polynomials containing any number of

terms.

To raise any polynomial to a power, we write it under the form of a binomial, each term of which may be a binomial, or some other polynomial; we then form the powers of these parts (by known methods), and proced exactly as with a true binomial.

(a+b+c)3

a3 + a2 + a

1 +(b+c)+(b2+2bc+c2)+(b3+3b2c+3bc2+c3)

a3+3a2b+3a2c+3ab2+6abc+3ac2+b3+3b'c

2. Find the square of (2a — x) + (3y + 4z).

+3bc2+3

1

+(3y+4z)+(9y2+24yz+1622)

(2a-x+3y+42)2 = 4a2-4ax+x2+12ay+16az-6xy-8x2

+9y2+24yz+1622

If a polynomial contains 5 terms, we first divide it into two parts, a binomial and a trinomial; find the powers of the trinomial, as shown above, and then proceed as indicated, and similarly for any polynomial what

11. (203 + 2x2 + 3x + 4)2 = 26 + 4x5 + 10x1 + 20x3 +

25x2+24x + 16.

Lime.

107. A Roor of a quantity is one of its equal factors, If a quantity be resolved into 2 equal factors, one of them is called the square root: the symbol for the square root is. If a quantity be resolved into 8 equal factors, one of these factors is called the cube root: the symbol for the cube root is 3. If a quantity be resolved into n equal factors, one of these factors is called the nth root: the symbol for the nth root is

.

108. The operation of finding one of the n equal factors of a quantity (n being any positive whole num ber whatever), is called extracting its nth root.

3

Instead of employing the symbols √, S, √, to indicate the square, cube, nth roots, it is more conve nient to employ the fractional exponents,, 1, *. which indicate the same thing. Thus,

It will be shown hereafter, that quantities affected with fractional exponents may be operated upon by the same rules as when they are affected with entire exponents