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 A reason for this may be seen if we write these terms in the form Example 1. Multiply 3x +x+2x3 by x3 – 2. Example 2. Divide 16a-3-6a-2+5a-1+6 by 1+2a-1. 2a-1+1)16a-3 - 6a2+ 5a-1+6 (8a-2-7a ̄1+6 16a-3+8a-2 Getting rid of the radical signs, and arranging in descending powers of x, we have 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. Multiply 32 −5+8x3 by 4x3+3x3. 1 2. Multiply 3a-4a-a ̈¦ by 3a3+a ̈3_ca Ca 3. Find the product of c*+2c-*-7 and 5-3c ̄*+2c*. 4. Find the product of 5 + 2,2a +3x-2a and 4.xa — 3x ̄a. 5a+4. 6. Divide 15a-3a3 — 2a 3+Sa-1 by 5a3+4. 7. Divide 16a-3+6a-2+5a-1-6 by 2a-1-1. 9. Divide 21a3* + 20 - 27a* - 26a2* by 3a* - 5. 10. Divide 8c ̄n − 8c2+5c3n − 3c-3n by 5c – 3c ̄". 12. 25a3+16-30a - 24a3+49a3. 13. 4x+9x ̄*+28-242-16.x. 14. 12a+4 −6a3x+a4x+5a2x. 3 3 1 - 15. Multiply a—8a2+4a3–2aa3 by 4a+a+4a3. 10 5 20. Divide 4x2-8a3 −5+3x + 3x 3 by 2x-12/x 22. 4/23-12 √(xy)+25y-24 √(3) + 23. 81 x-2 24. 2 .2.3 (+1) +1+ Ny 16 Ny-2 2x 3 +16x3y. x2 y 248. The following examples will illustrate the formula of earlier chapters when applied to expressions involving fractional and negative indices. Example 2. Multiply 2x2P - x2+3 by 2x2 + x2 --- 3. 1 1 1 = 9x + 4 + x ̄1 − 2 . 3x2. 2 -2.3.x.x |