The series of powers, by division, may be extended backward. Thus, if we divide a5/a= a 4; a4/a = a3; a3/a = a2; a2/a= a1;a1/a = a0; a0/a= a-1; a-1/a = a-2; a-2/a = a-3, etc. In this series we have a/a = a0 But a quantity divided by its equal gives unity for quotient, or 3/3 = 1. Therefore, a/a = 1, and ao - 1. This result is remarkable, and holds good regardless of the value of a.

From this and the preceding negative exponents we derive the following:

ao = a/a = 1, a-1 = ao/a = 1/a,

a-2= a-1/a = 1/aa = 1/a2

a -2 =a-1/a = 1/aa = 1/a2;

a3=a-2/a =1/a2a = 1/a3,etc.

Showing that a quantity with a negative exponent may have substituted for it the same quantity with a positive exponent, but used as a denominator to a fraction having unity for the numerator.

423. - Addition And Subtraction Of Exponential Quantities

Equal quantities raised to the same power may be added or subtracted; as, a2 + 2a2 = 3a2; but expressions in which the powers differ cannot be reduced; thus, a2 + a - a3 cannot be condensed.

424. - Multiplication Of Exponential Quantities

It will be observed in Art. 421 that in the series of powers, the index or exponent increases by unity; thus, a1 a2 a3 a4 etc.; and that this increase is effected by multiplying by the root, or original quantity. From this we learn that to multiply two quantities having equal roots we simply add their exponents. Thus the product of a, a2 and a3 is a1 x a2 x a3 = a6 The product of a2, a3, and a5 is a-2 x a3 x a5 = a6. The exponents here, are: - 2 + 3 + 5 = 8 = 2 = 6.

425. - Division Of Exponential Quantities

As division is the reverse of multiplication, to divide equal quantities raised to various powers, we need simply to subtract the exponent of the divisor from that of the dividend. Thus, to divide a5 by a2 we have a5-2 = a3. That this is correct is manifest; for the two factors, a2 x a3, in their product, a5 produce the dividend.

To divide a2 by a5 we have a2-5 =a-3, which is equal to 1/a3 (see Art. 422). The same result may be had by stating the question in the usual form. Thus, to divide a2 by a5 we have a2/a5, a fraction which is not in the lowest terms, for it may be put thus, a2/a2a3= a2 /a5 by which it is seen that it has in both its numerator and denominator the quantity a2 which cancel each other (Art. 371). Therefore, a2/a5 = 1/a2; the same result as before.