Part 14

Review the proofs, for positive integral exponents, of:

I. a^m x a^n = a^(m + n).

II. (a^m)/(a^n) = a^(m - n).

III. (a^m)^n = a^(mn).

IV. [a^(mn)]^(1/n) = a^m.

V. [a/b]^n = (a^n)/(b^n).

VI. (abc)^n = a^n b^n c^n.

~To find the meaning of a fractional exponent.~

Assume that Law I holds for all exponents.

If so, a^(2/3) . a^(2/3) . a^(2/3) = a^(6/3) = a^2.

Hence, a^(2/3) is one of the three equal factors (hence the cube root) of a^2.

Therefore a^(2/3) = [a^2]^(1/3).

In the same way,

a^(4/5) . a^(4/5) . a^(4/5) . a^(4/5) . a^(4/5) = a^(20/5) = a^4.

Hence, a^(4/5) is one of the five equal factors (hence the fifth root) of a^4.

Therefore a^(4/5) = [a^4]^(1/5).

In the same way, in general, a^(p/q) = [a^p]^(1/q).

Hence, the numerator of a fractional exponent indicates the power, the denominator indicates the root.

~To find the meaning of a zero exponent.~

Assume that Law II holds for all exponents.

If so, (a^m)/(a^m) = a^(m - m) = a^0. But by division, (a^m)/(a^m) = 1.

Therefore a^0 = 1. Axiom I.

~To find the meaning of a negative exponent.~

Assume that Law I holds for all exponents.

If so, a^m x a^(-m) = a^(m - m) = a^0 = 1.

Hence, a^m x a^(-m) = 1.

Therefore a^(-m) = 1/(a^m).

Rules: