THE IKKIGATION AGE. 



743 



If two powers of the same base are divided into each other, 

 the resulting quotient can be simplified, for instance : 



3 2 



w 



F 



3* 



?i 



3 



3X3X3X3X3 



3X3 

 3X3X3X3X3 = 



3X3X3 = 3 3 = 3 s - 3 



3X3X3 

 3X3X3X3X3 



3X3X3X3 

 3X3X3X3X3 

 3X3X3X3X3 



3X3X3X3X3 

 3X3X3X3X3X3 

 1 3 5 - 7 



= * = y = 35-3 



= 3 1 = * 3 5 ~ 4 



1 1 5 _ 6 



3 3' 



From the above it appears that powers of the same base 

 are divided into each other by subtracting the exponent of 

 the divisor from the exponept of the dividend. Also, that 

 a power having the exponent zero equals 1 ; also, that a power 

 with a negative exponent represents a fraction with the 

 numerator 1 and the denominator is the power with positive 

 exponent. Illustrating the same principles upon algebraic 

 terms we have : 



^ = a 



= a" = 1 



nJO 



d a lS 17 _ a 4 



a 17 a 4 



If two powers of the same base are multiplied together 

 it is done by adding the exponents together, thus : A 3 XA 4 = 

 A T , or (B+C) 5 X(B+C) 8 =(B+C) 13 . 



If, however, a power is to be raised to another power 

 like X 5 is to raised to the third power then it is indicated 

 thus: (X 5 ) 3 =X 15 , which means that the exponents must be 

 multiplied. 



NOTE: The second power of a number is called the 

 square, and the third power is called the cube of such num- 

 ber. 



7. Evolution. In the expression : 4 3 =64 are three quanti- 

 ties, the base 4, the exponent 3 and the power 64, and if any 

 two of them are given, the third one may be found. Involu- 

 tion showed how the power is found when the base and the ex- 

 ponent are given. Thus, the power 64 in the above term is 

 found by multiplying the base 4 three times by itself. Sup- 

 pose, however, the power 64 and exponent 3 are given and it 

 is required to find the base 4, then the problem is to split 

 the power 64 into as many equal factors as the exponent 3 

 indicates. This operation is called evolution and is indi- 

 cated thus : 



X = 



Read : X=the third root of 64 ; X is now called the 

 root, 3 is the index, 64 is the radical and the sign between 

 3 and 64 is called the radical sign. If the index is 2 it is 

 called the square root, and is usually omitted; if the index 

 is 3 it is called the cube root ; if the index is 5 it is called the 

 fifth root, etc. In solving the various problems in applied 

 hydraulics it is occasionally necessary to extract the square 

 root, cube root, fifth and seventh root. The extraction of 

 the square root is a familiar operation with many people, 

 but the cube root is much more difficult, and higher roots 

 are usually extracted by the use of logarithms. Tables of 

 square roots and cube roots are appended, especially selected 

 for the use in hydraulic work and the following paragraph 



gives hints, how to find any root by the use of logarithms. 

 8. Logarithms. In 7 it was stated that powers of the same 

 base are multiplied together by adding the exponents, that 

 powers of the same base are divided into each other by 

 subtracting the exponents, that powers are raised to another 

 power by multiplying the exponent by the other power; in- 

 versely the root may be extracted from any power by dividing 

 the exponent of the oower by the index of the root. To illus- 

 trate these principles briefly, we will use the base 10; then: 



10- X 10" = 10 2 + :1 = 10 s 

 10" * 10 2 10 6 - 2 = 10 4 



(10 5 ) :! : 105X3 = iQi-' 



1/10* = 10 7 



Every number can be considered as a power of 10 ; thus : 



1=10. 



10=10'. 



100= 10 2 . 



1,000=10 3 . 



10,000=10*. 



100,000=10. 



1,000,000=10'. 



Also for number smaller than 1 : 

 .1=10-'. 



.01-10-". 



.001=10- 3 . 



.0001=10-*. 



.00001=10- 5 . 



.000001=10-. 



These exponents of the base 10 producing the different 

 numbers are called logarithms. Thus, it is seen the logarithm 

 of 1 = 0, the logarithm of 10=1, of 100=2, of 1,000=3, of 

 10,000=4, etc. So the logarithm of all numbers between 1 

 and 10 are greater than and smaller than 1; the logarithms 

 'of all numbers between 10 and 100 are greater than 1 and 

 smaller than 2; the logarithms for all numbers between 100 

 and 1,000 are greater than 2 and smaller than 3 ; it shows 

 also that all logarithms are decimal fractions, except the 

 exact powers shown. 



Tables of these logarithms have been computed and a 

 table of them is embodied into this book. 



9. Formulas. A formula is an algebraic expression show- 

 ing how to apply a scientific principle to the solution of a 

 practical problem. They are always in the form of an 

 equation and most of them are simple enough to be handled 

 coPfectly by anyone having a common school education. As 

 an illustration let the box 



A, B, C, D, E, F, G, H, shown in Fig. 3 be of a rectangular 

 shape; if the depth AD=3 feet, the width DC=4 feet, and 

 the length CE=6 feet, then the volume enclosed by the box 

 would be 3X4X 6=1 '' 2 cubic feet, and as a cubic foot of water 



