EXERCISES II.

Simplify each of the following expressions:

1. (2+4)+(2 i — 3).

2. (7-5)-(3-4 i).

3. (1+√−9)+(4−√−4). 4. (6−√−16)–(5−√−36).

9. (3+5 i) (√12 – 3 i).

6. (2+i√3) (2 — i√3).

7. (2+3√−1)(3—4√−1). 8. (7 + √−5) (7 −√−5). 10. (√8−√−12)(√2−√−3). 11. (‡ −‡i√3)(3+3i√3). 12. (5 −2 i√√6) (5+2 i√6). 13. [x+i√(a − x2)][x — i√(a — x2)].

Make the indicated substitution in each of the following expressions, and simplify the results:

CHAPTER XVII.

DOCTRINE OF EXPONENTS.

1. We have already abbreviated such products as

ад, aдa, aдаа, ..., ааа ... n factors,

by a2, a3, a, ..., a", respectively, and called them the second, third, fourth, ..., nth, powers of a. This definition of the symbol a requires the exponent n to be a positive integer.

Thus 25 means the product of 5 factors, each equal to 2. But 20 has, as yet, no meaning, since 2 cannot be taken 0 times as a factor. For a similar reason 2-5 and 2a are, as yet, meaningless.

But, having introduced into Algebra the symbol a", it is natural to inquire what it may mean when ʼn is 0, negative, or a fraction.

We shall find that, by enlarging our conception of powers, quite clear and definite meanings can be given to such expressions as 20, 3-2, and 4.

2. The principle

Positive Integral Powers.

am xan am+n,

=

wherein m and n are positive integers, was illustrated by particular examples in Ch. III., Art. 24.

In general,

am xan= (aaa to m factors) (aaa... to n factors)

...

3. The other principles upon which operations with positive integral powers depend have been proved in the preceding chapters.

For the sake of emphasis, and for convenience of reference,

4. The meaning of a symbol may be defined by assuming that it stands for the result of a definite operation, as was done in letting

an = a • a • a • ... n factors;

or by enlarging the meaning of some operation or law which was previously restricted in its application.

In the latter way, negative numbers were introduced by extending the meaning of subtraction.

5. We now enlarge the meaning of powers by assuming that the principle

That is, the zeroth power of any base, except 0, is equal to 1.

E.g.,

1o = 1, 5o = 1, 99o = 1, (a + b)° = 1, etc.

6. Thus, by the assumption that the stated law holds when m=n, a definite value of the zeroth power of a number is obtained. Nevertheless, it will doubtless seem strange to the student that all numbers to the zeroth power have one and the same value, namely 1. But it should be distinctly noted that

am

a is by definition a symbol for ; i.e., for the quotient of two like powers of the same base. Thus,

7. We now still further enlarge the meaning of powers by assuming that the principle

holds not only when m>n and m = n, but also when m<n. We then have, for example,

In general, since m < n, we may assume n = m + k.

That is, a negative power of a number is equal to the reciprocal of a positive power of the same number, the exponents being numerically equal.

This relation and the relation which defined a negative integral power may be stated thus:

Any power of a number may be transferred from the denominator to the numerator, or from the numerator to the denominator, of a fraction, if the sign of its exponent be reversed.

This reciprocal relation between positive and negative powers is useful in reductions which involve negative powers.

Change each of the following expressions into an equivalent expression in which all the exponents are positive:

In each of the following expressions transfer the factors from the denominator to the numerator: