CHAPTER VIII POWERS AND ROOTS POWERS 354a. 2 x2=4; 2x2x2 = 8; 2 × 2 × 2 × 2 = 16. 3 x 39; 3 x 3 x 3 = 27. In problem 1, how was 4 obtained? 8? 16? In problem 2, how was 9 obtained? 27? Note that in each case the product was obtained by multiplying together equal factors. A product, or number produced by multiplying a number by itself, is called a power of that number. The number itself is called the first power. The product resulting from multiplying the number by itself once is called the second power or square; that resulting from multiplying the number by itself twice, the third power or cube; 4 is the second power of 2, and 8 is the third power of 2. What is the second power of 3? Of 4? Of 6? Of 8? What is the third power of 3? Of 4? Of 6? Of 8? 2 × 2 × 2 may be written 23. 3 x 3 may be written 32. 5 × 5 × 5 × 5 may be written 54. The small figures, 3, 2, and 4, are called exponents. An exponent shows how many times the number is used as a factor. It is written at the right of the number and a little above it. 434 x 4 x 4. What does 35 equal? 68? 72? 83? 94? 355. First power POWERS OF NUMBERS TO 9 = 1, 2, Second power = 1, 4, 3, 4, 5, 6, 7, 8, 9. 9, 16, 25, 36, 49, 64, 81. Third power = 1, 8, 27, 64, 125, 216, 343, 512, 729. Study this table until you can readily give the second and the third powers of the numbers in the first row. What number multiplied by itself will produce 36? ROOTS 356. What are the equal factors of 36? Of 49? Of 27? Of 8? Of 216? Of 64? A number which is multiplied by itself to produce a given number, or which is one of the equal factors of the number, is called a root of that number. If the root is one of two equal factors, it is called the square root; if it is one of three equal factors, it is called the cube root. 357. In the problem, 4 = 2 × 2, 2 is the square root of 4, because it is one of the two equal factors. 359. In the problem, 82 × 2 × 2, 2 is one of the three equal factors and is the cube root of 8. Find the cube root of the following numbers by finding one of their three equal factors: 343, 216, 1,000, 1,728, 2,744, 512. 360. Instead of writing the words square root, cube root, etc., mathematicians use a sign to indicate roots. They use the sign √,, . It is called the radical, or root sign. When it is used without the figure at the opening, the square root is indicated; when the figure 3 is used, the cube root is indicated. In arithmetic, roots above the cube root are not considered. √9 indicates the square root of 9. 64 indicates the cube root of 64. 361. The following method enables one, in many cases, to determine at a glance the square root of a number. It grows out of the method of multiplying in which every complete product is put down, instead of adding and combining as one multiplies. NOTE TO TEACHERS: Two methods of finding square root are here given, the "Multiplication method" and the should select one and teach that alone, "Geometric method." Teachers not both. FINDING ROOTS THE MULTIPLICATION METHOD 362. These multiplications teach us how to proceed without factoring if we wish to find the square root of 144, or the number which multiplied by itself makes 144. It will be seen that 144 is made up of two squares, 102 + 22, and twice the product of the numbers making the squares, 2 x 10 x 2. It can be separated into these parts as follows: 363. Find the square root of the following numbers by the method just given: 1. 361. 2. 729. 3. 7,225. 4. 5,929. 5. 1,296. 6. 1,681. 9, the largest number expressed by one figure, when squared, gives 81, a number of two figures. 10, the smallest number expressed by two figures, when squared, gives a number of three figures; and 99, the largest number expressed by two figures, when squared, gives 9,601, a number of four figures. 100, the smallest number expressed by three figures, when squared, gives 10,000, a number of five figures; and 999, the largest number expressed by three figures, when squared, gives 998,001, a number of six figures. Therefore, if the number of which the square root is to be found contains one or two figures, there will be one figure in the root; if it contains three or four figures, there will be two figures in the root; if it contains five or six figures, there will be three figures in the root. ORAL EXERCISES 365. How many figures will there be in the square root of each of the following numbers? 366. The first step in extracting the square root of a whole number is to divide it into periods of two figures each, beginning at the right. In the numbers that you have produced by squaring other numbers, you will observe that the tens squared can always |