35.

2n X (2n-1) 1

2n +1

4n+1 24 + 1 x 21-1* 4

36.

(2n-1)+10 246. Since the index-laws are universally true, all the ordinary operations of multiplication, division, involution and evolution are applicable to expressions which contain fractional and negative indices.

247. In Art. 182, we pointed out that the descending powers of x are

1 1 1

23, 2°, x, 1, A reason for this may be seen if we write these terms in the form

23, x2, X1, 2:0, x-1, 3-2, 2-3, ......

x2

1

1

Example 1. Multiply 3.x* +x+ 2x3 by x3 – 2.
Arrange in descending powers of X.

2

1
20 +2.03 + 3.0 3

1
x3 - 2

Example 2. Divide 16a-3 6a-3 + 5a-1 +6 by 1+ 2a-1. 2a-1 +1)16a-3 - 6a-+ 5a-1 +6(8a-2 – 7a-1+6

16a3+ 8a-2

- 14a--2+ 5a-T
- 14a-2- Ta-1

12a-1+6
12a-1 +6.

4.ro

+

Example 3. Find the square root of

- 2.c +*+x3 – 4 (x8y-?). y

4

y Getting rid of the radical signs, and arranging in descending powers of x, we have

1 2

NOTE. In this example it should be observed that the introduction of negative indices enables us to avoid the use of algebraical fractions.

EXAMPLES XXXI. c.

1

1

1

1. Multiply 3x2 – 5+8x 3 by 4.x2 + 3x 3. _57823

4x8+3.01 2. Multiply 3ad-tad_aby zabta -ca 3. Find the product of cut + 20% – 7 and 5 – 3c-*+2c*. 4. Find the product of 5 + 2.3.2a +3.x~2a and 4.ra – 3.x-a. 5. Divide 21x+23+.23 +1 by 3.23 +1.

. 6. Divide 15a - 303 – 2a 3 +8a-1 by 5a' +4.

– +. 7. Divide 16a-3 +6a-?+5a-1-6 by 2a-1-1. 8. Divide blå_643 – 40 3 – 46 3-5 by iš -26 9. Divide 21a3x + 20 – 27a? – 26a2x by 30% – 7. 10. Divide 8c-n – 80n + 5c3n – 3c-3n by 5cm – 3c-n.

Find the square root of 11. Ix – 12.c+10 – 4x +x-1, 12. 25a3+ 16 – 30a - 24a3+49a3. 13. 4.x" + 9x-n + 28 – 24x

.

- 16.02. 14. 12ax +4-6a3x + 24x +5a-t. 15. Multiply a _ 8a +4_ Lał by sa tał + 4a

2

1

16. Multiply 1-2/x – 2x2 by 1 - Ix.

17. Multiply 24a6_a_3 by 2a-3

Vaat 18. Dirido 2F+0/-10% : by Atat

4

2 19. Divide 1-va-. i

20. Divide 4 1.33 – 8 5+ + 3x š by 2018_1.x

3

1-3 NY --6 Ny.

22

+

Vy-2

2x

. a 2 +(a

Find the square root of

15y 21. 9x-4 -- 18x^3 Jy+ 6 22. 41.23 -- 12 V(x?y)+25 Ny – 24 +16x ay.

X 23. 81 -+1) +36 (.x23 y-1-1) - 158 Ny

y 9 24. +1+

16 248. The following examples will illustrate the formulæ of earlier chapters when applied to expressions involving fractional and negative indices. Example 1. (ał- bé) (a1 + 0 %) =*-*- a två tako smo

h P

P
=1-a bi tako 9-1

=ukl - a k b9. Example 2. Multiply 2x2P – XP +3 by 2x2P + XP - 3. The product = {2x2P – (XP - 3)} {2x2P + (xP – 3)}

= (2x2P)2 – («<P – 3)2

=4x4P – x2P+CXP – 9. Example 3. The square of 3.2-2-2 =9x +4+x-1- 2.3.x2.2 - 2.3.x2.x

-2.3.27.2-2.3.27.2 12.2.41 =9x +4+x-1 - 12x2 - 6 + 4x 2

. =9x – 12.x2 - 2 + 4x2+x-1, by collecting like terms and rearranging. Example 4. Divide a? +a

2 by až + a The quotient =(a2 +a )=(a2 + a

= {(4%)* + (a**)3} = (að+a )

– až + ?) =an - 1+a-n.

1

1

1

1

1

3n

3n

n

3n

3n

n

n

2

EXAMPLES XXXI. d. Write down the value of 1. (cd-7)(**+3). 2. (4x – 5x-1)(4x +3.c-1). 3. (7x - 9y)(7x+9y-1). 4. (.com - y")(x-+y"). 5. (a* - 2a-*) 6. (a*+az). 7. 8. (5.217° - 3.r-ay-)(4.2"yb +5x-ay-)).

1 9.

10. (3.ray-b+5x-ay)) (3.rayb – 5.x-"y-b). 11. (-) 12. (222 – x a + x)2.

+ 13. (a+b)x+(a - b) va 14. {(a+b)2 – (a - b) ??

Write down the quotient of

1\2

1

1

25. (x+4)(x+x} + 4). 26. (2xé+4+3x+5)(2x:+4–32. ) 27. (2_x3+x)(2+23+x). 28. (a+7+3a-=)(q–7–3a-3).

+ (1+2)" að+2 Jab+463 203 – 43x-2 9.x2+4+4.00

+1

29.

30.

2

31.

32.

a2 + ab va
ab – 63 sub