Plane trigonometry with practical applications, by Leonard E. Dickson.

CHAPTER IV
LOGARITHMS, SLIDE RULE
23. Powers of 10. The symbol 102 denotes the square of 10
and equals 10 X 10 = 100. Again, 103 denotes the cube of 10 and
equals 10 X 10 X 10 = 1000. The product 102 X 103, being the product of two factors 10 by three factors 10, equals 105, which is the
product of five factors 10. It follows that the quotient of 105 by 102
equals 103. These facts are expressed by the following equations:
102 X 103 = 105, 105 - 102 = 103.
Similarly, if m and n are any two positive whole numbers,
(1) 1om X 10n = 0 + n,
(2) 10m -+ 10 = 10o- (if m>n).
Just as (103)2 = 103 X 103 = 106, so always
(3) (10m)n = 10n.
Formulas (1), (2), (3) are known as the first, second, and third
index laws, respectively.
We next inquire what meaning must be given to symbols like
10, 10%, 10-2 if also these symbols shall obey the index laws.
Using the third law (3), we have
n
(101)2 = 101 = 10, (10))3 = 10, (10d)d= 10=
so that 10- is the (positive) square root of 10, 103 is the cube root
of 10, and, in general, 10l/d is the dth root of 107:
n
(4) 10 = a/10, 103 =, 10. =
We agree to employ the second law (2) also in the new case
m = n = 2, whence
102 - 102 = 100.
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