HARVARD COPYRIGHT 1912 BY AMERICAN SCHOOL OF CORRESPONDENCE Entered at Stationers' Hall, London All Rights Reserved PRACTICAL MATHEMATICS PART III POWERS AND ROOTS 59. Powers. A power of a quantity is the product of factors, each of which is equal to that number. This quantity may be simply a number, as 1, 2, 3, or 4, or it may be any letter, as a, b, c, or d, which may have any numerical value whatso ever. In order to show how many times the quantity is to be used as a factor, a small figure is placed to the right and a little above the quantity; as 3 or a1. This small number is called an exponent, and shows to what power the quantity is to be raised. Thus, 34 means the fourth power of 3, or 3 × 3 × 3 × 3 = 81; i. e., 3 is taken 4 times as a factor. If the letter a is substituted for the figure 3 and this letter is raised to the fourth power, the result is a Xa X ax a or a1. Since no definite value for a has been given, the result of raising the letter to the power can only be indicated; thus, a1. X the cube of a. The second power of the quantity is called its square. For example, 4 is the square of 2, for 2 X 2 = 4; again, a2 is the square of a, for a × a = a2. The third power of a quantity is called its cube; thus, 8 is the cube of 2, for 2 X 2 X 2 = 8; or a3 is Suppose now it is required to find the square of a2. for a2 X a2 = (a × a) × (a × a) = a; or, in other two like quantities are multiplied together, the exponent of the product is equal to the sum of the exponents of the quantities. a2 × a2 The power of a fraction is obtained by multiplying the numerator by itself and the denominator by itself the required number of times. Examples. 1. What is the cube, or third power of 21? SOLUTION. Here it is readily seen that raising 21 to the third power is the same as using 21 three times as a factor. 2. What is the cube, or third power of 2a? SOLUTION. Note here, that when raising a quantity like 2a to a power, the coefficient 2 and the letter a are each raised separately to the required power. PROBLEMS FOR PRACTICE Raise the following quantities to the power indicated: 60. Roots. A root of a quantity is one of the equal factors which, when multiplied together, give the quantity. Thus, if a certain quantity is used twice in order to produce another quantity, then the quantity first mentioned is the square root of the second. Thus, 2 is the square root of 4, for 2 X 2 = 4. If the first quantity must be used three times as a factor, it is the cube root of the second quantity; thus, the cube root of 8 is 2, for 2 X 2 X 2 = 8; the cube root of 27 is 3, for 3 × 3 × 3 = 27. This also applies to any root as the 4th, 5th, etc. Thus, the 5th root of a = a, for a Xa Xa xa xa This process of finding the root is merely the reverse of finding the power, and is termed extracting the root. The radical sign√when placed before a quantity shows that some root of it is to be taken. The root is indicated by a small figure called the index placed above the radical sign; thus,√. When no index is written, the square root is always understood. The following examples show the meaning of the sign and index: |