28. (3 a-10+a6 − 4 a ̃o) ÷ (2 a−2 + a2 + 3 a−o).

29. (2 x3-3x2 - 2 x1 +2 - x) + (x−1 + 1).

30. (x−1 − 3 x ̄ + 3 − 3 x1 − 2 x) ÷ (x ̄‡ − 2 x−1 + x ̄1 − 2).

31. (2a-3 a3-23 a1 +15 a-5 +9 a ̄) ÷ (a* + 2-3 a-1).

32. (6 +92–2 -_13)+(3

+2–5).

34. (x*+a3x3 +a3) + (x2 + a1x1 + a3).

35. (a§ —a2b1 — a‡b2 + b3) ÷ (aa — ab1 +a‡b −b}).

36. (6 * _72_19 – –
19 x2 + 2 x1 + 8 x1) ÷ (2 x‡ − 3 x1 − 4 x1).

(III.)

x -7x

(a")" = amn,

Ex. 3. (a ̄3‚x2y−1) ̄‡= (a ̄3) ̄}(x) ̃4(y−1)−4 = a$x¬1y? — a‡y2

(i.) m and n both negative integers.

=

Let m =- m1 and n = - n1, so that my and n1 are positive.

In a similar manner the principle can be proved for other cases in which the exponents are 0 or negative integers.

(ii.) m a fraction, and n a positive or a negative integer, or 0.

In a similar manner the principle can be proved when m is an integer and n is a fraction.

(iii.) m and n both fractions.

Let m= P and n = wherein and s are positive integers, and p and

q

-9

8

r are positive or negative integers.

Pr

If (a) be raised to the qsth, = sqth power, we have

Consequently (a) is the qs root of arr; or, by definition of a fractional power,

(IV.) (ab)m = ambm, for all rational values of m.

(i.) m a negative integer. Let m = - m1, so that m1 is positive. Then

(ab)m = (ab)-m1

1

=

(ab)m

1

ambm1

= amib¬mi = ambm.

(ii.) m a fraction.

wherein p is a positive or negative integer, and q is a posi

tive integer.

22. An Irrational Power is a power whose exponent is an irrational number; as xv2.

23. Let a be any real positive number greater than 1, and I be a positive irrational number defined by the relation (Ch. XVII., Art. 6):.

Then the two rational powers a" and a " have the properties (i.) and

(ii.), Art. 6, Ch. XVII.

a" increases and a "decreases as n increases [Art. 19 (i.)], and an <an.

m+1

The difference a 78

ana"(an - 1) is positive and can be made less than any assigned number, however small.

always less than some positive finite number, say R.

m

m+1

26

therefore an is

Therefore the given difference can be made less than Rd. But Rd can be made less than any assigned number, say 8, by taking d less than

Therefore the two series of powers a" and a define a positive number which lies between them. This number is defined as a1.

n

That is,

In the proofs of the principles which follow we shall assume that the base is greater than 1.

25. It follows directly from the definition of a-1, that

26. It can now be proved that the principles of rational powers hold

also for irrational powers.

Let a4 and a be two irrational powers defined by the relations