241. To find the meaning of zero exponent, e.g. ao.

The operation which makes the zero exponent disappear is evidently a multiplication by any power of a, e.g. a2.

Therefore the zero power of any number is equal to unity.

a2

NOTE. If, however, the base is zero, indeterminate.

a2

is indeterminate; hence 0° is

242. To find the meaning of a negative exponent, e.g. a-".

Multiplying both numbers by a", a"x = ao.

243. Factors may be transferred from the numerator to the denominator of a fraction, or vice versa, by changing the sign of the exponent.

1 n

1. a

NOTE. The fact that a = 1 sometimes appears peculiar to beginners. It loses its singularity if we consider the following equations, in which each is obtained from the preceding one by dividing both numbers by a. a3=1.a. a⋅ a

THE LAWS FOR NEGATIVE AND FRACTIONAL

EXPONENTS

244. Exponent law of division for any values of m and n.

To prove aman am-n for any value of m and n.

Hence the law is true for any values of m and n.

(§ 243)

(§ 239)

245. Exponent law of involution for any values of m and n.

To prove (am)n = amn for any values of m and n.

Case 1. Let m have any value, and `n be a positive integer. This was proved in a preceding chapter (§ 239).

Case 2. Let m have any value, and n be a positive fraction

(§ 240) (Case 1)

(§ 240)

Case 3. Let m be any number and n be negative.

246. In a similar manner it can be proved that the law (ab) = ambm is true for fractional and negative exponents.

Hence the four laws of exponents are true for any value of the exponents, and we have, in general,*

Fractional and negative exponents are treated by the same methods as positive integral exponents.

247. Examples relating to roots can be reduced to examples containing fractional exponents.

Ex. 1. (alb-1) ÷ (alv−1)} = atb−1 ÷ alb−1 = a ̄to}

248. Expressions containing radicals should be simplified as follows:

(a) Write all radical signs as fractional exponents.

(b) Perform the operation indicated.

(c) Remove the negative exponents.

(d) If required, remove the fractional exponents.

NOTE. Negative exponents should not be removed until all operations of multiplication, division, etc., are performed.

* Irrational and imaginary exponents cannot be considered at this stage of the work.