17.

25. A warehouse will hold 24 boxes and 20 bales. 6 boxes and 14 bales will fill half of it. How many of each alone will it hold?

26. If the greater of two numbers is divided by the smaller, the quotient is 7 and the remainder 4; but if 3 times the greater is divided by twice the smaller, the quotient is 11 and the remainder 4. Find the numbers.

27. If the sides of a rectangular field were each increased by 2 yards, the area would be increased by 220 square yards. If the length were increased and the breadth diminished, each by 5 yards, the area would be diminished by 185 square yards. Find the dimensions of the field.

28. The report of an explosion traveled 1039 feet per second against the wind and 1061 feet per second with the wind. Find the velocity of sound in still air and the velocity of the wind.

29. If 5 is added to both numerator and denominator of a fraction its value is. If 3 is subtracted from both, its value is. Find the fraction.

30. If two trains 100 miles apart, approach each other, they will meet in 2 hours; but if they run in the same direction, the slow one leading, they will be together in 10 hours. Find the rate of each train.

31. Find 3 numbers such that if each be added to the sum of the others, the results will be 32, 28, and 30, respectively.

32. It is known that the sum of the three angles of a triangle is 180°. The difference between the first and second angles is 10°, and between the second and third angles is 25°. Find the number of degrees in each angle.

33. A certain number of two digits equals six times the sum of its digits, and if the digits were interchanged, the resulting number would be 9 less than the given number. Find the number.

34. A tank has two pumps. If the first is worked 2 hours and the second 3 hours, 1020 cubic feet of water are discharged. But if the first is worked 1 hour and the second 2 hours, 690 cubic feet are discharged. How many cubic feet can each pump discharge in 1 hour?

CHAPTER XIII

THEORY OF EXPONENTS

101. The rules for the exponents of like letters have been explained for multiplication and division. Thus, to multiply a2 by a3, the exponent of a in the product will be the sum of the exponents 2 and 3 of the factors, that is the product will be a5. Similarly, in dividing a5 by a3, the exponent 3 of the divisor is subtracted from 5, the exponent of the dividend, and the quotient will be a2.

The operation of raising any quantity to a required power is called Involution.

Thus, to find the cube of a2, we must find the product of three factors, each equal to a2. As each factor contains two factors a, the product will contain six factors a; hence the cube of a2 equals a, that is, the cube of a quantity contains three times as many equal factors as the quantity itself. Similarly, the fifth power would contain five times as many factors. In general, therefore, any required power of a quantity can be found by multiplying the exponent of that quantity by the exponent of the power to which it is to be raised.

This rule will hold whether the exponents are numbers or letters.

If the quantity consists of two or more unequal factors, the rule applies to each factor.

The operation of finding any required root of a quantity is called Evolution.

Thus, the cube root of a is one of the three equal factors

root of a will contain

of a. Since a contains 6 factors each equal to a, the cube of the total number of six factors, that is, two a factors. Hence the cube root of a equals a2. To indicate a root, the root sign or radical sign, √, is used, with the proper index placed within the sign. Thus the cube root of a is written a®.

The rule of exponents for evolution is, therefore,

Any required root of a quantity can be found by dividing the exponent of the quantity by the index of the root.

102. It is evident that of the six operations, addition, subtraction, multiplication, division, involution and evolution, the only two that can not be performed by means of the exponents, are addition and subtraction.

Multiplication is performed by addition of exponents.
Division is performed by subtraction of exponents.
Involution is performed by multiplication of exponents.
Evolution is performed by division of exponents.
The rules for exponents may be briefly stated:

The last four of the six operations are performed by the first four of the exponents.

103. The exponents considered, so far, have been positive integers. In applying the rules for exponents, other forms of exponents frequently arise.

Meaning of a zero exponent.

Whenever a number is subtracted from its equal, the result is zero.

A zero exponent, therefore, is produced whenever an exponent is subtracted from an equal exponent. But subtraction of exponents occurs in division. Hence, whenever a quantity is divided by its equal, the exponent of the quotient will be zero.

its equal will yield a quotient equal to 1.

A zero exponent therefore, indicates that a quantity has been divided by its equal. Any quantity with a zero exponent is equal to 1.