CHAPTER X SQUARE ROOTS AND TABLES 67. Powers. In taking up the calculation of square roots we must first come to an understanding of the terms "powers" and "roots." A number is said to be "raised to a power" when it is multiplied by itself a certain number of times. The number of times which it is multiplied together determines to what power it is raised. Thus, 2 raised to the second power is 2 X 2 = 4; 2 raised to the third power is 2 X 2 X 2 = 8; 5 raised to the third power is 5 X 5 X 5 = 125; 3 raised to the fourth power is 3 × 3 × 3 × 3 = 81; etc. Instead of writing out the multiplications as 5 X 5 X 5 = 125, it is customary to write a small figure, called the "exponent," at the right of the number and slightly higher, which indicates the power it is raised to; thus, 53 125. The small 3 indicates the third power of 5. Likewise, instead of 3 X 3X 3X3 = 81, we would write 34 81, which is read "3 to the fourth power equals 81." = = The second power and the third power of any number are usually called the "square" and the "cube," respectively, and to raise a number to these powers, we "square" or "cube" the number. For instance, the square (or second power) of 6 is 62 36; the cube (or third power) of 4 is 43 = 64; 5 cubed is 53 = 125; 3 squared is 32 = 9, etc. = 68. Roots. Roots are the opposite of powers. The fourth power of 2 is 24 = 32, and conversely the fourth root of 32 is 2. Roots are designated as square roots, cube roots, fourth roots, fifth roots, etc., the same as powers. To find some root of a number, such as the cube root of 27, we find a number which taken as a factor three times will give a product of 27. Thus 3 is the cube root of 27, since 3 × 3 × 3 = 27; 5 is the fourth root of 625, since 5 X 5 X 5 X 5 = 625; 6 is the square root of 36, since 6 X 6 = 36. The square root of a given number is, therefore, another number which when multiplied by itself will produce the given number. The square root of 4 is 2, the square root of 9 is 3, the square root of 16 is 4, etc. Just as the small figure at the right is used to indicate the power of a number, so there is a conventional method of indicating roots. Taking the square root of a number is designated by the sign placed over the number, and this sign is called the "square root" or "radical" sign. Thus 42, which is read "the square root of 4 equals 2;" √9 3, etc. To represent a different root, such as the cube root, fourth root, etc., a small figure designating the root is placed at the upper left-hand corner of the radical sign. Thus, the cube root of 27 is written 27 = 3; the fourth root of 16 is written 16 2. = = 69. Extracting the Square Root.-Finding the square root of a given number is called extracting the square root. For simple numbers such as 4, 9, 16, 36, etc., the square root can be determined by inspection, but we must have some method that will apply to any number. The actual calculation of square roots is a somewhat laborious process, but not a difficult one to learn. The first step in finding the square root of a number is to divide the number into periods or groups of two figures each, beginning at the decimal point and working both ways. The following examples indicate how this is done. 9'20'. 2'53'20'. '76'30'.79'9 6'.42'97' The number of periods to the left of the decimal point thus marked off determines the number of digits to the left of the decimal place in the square root. Thus the number 10'24'. has two groups or periods and its square root, 32, has two digits to the left of the decimal place. The remainder of the calculation can best be explained by the solving of some examples, explaining them as we go along. Example: Find the square root of 186,624. Point off into periods of two figures each (18'66'24') as explained, and it will be seen that the root must contain three digits to the left of the decimal point. The work is arranged very similarly to division. Explanation: Find the largest integral number whose square is equal to, or less than, the first period, 18. This is 4, since 52 is more than 18. Write the 4 to the right for the first figure of the answer, just as the quotient is put down for division. Square the 4 and place its square, 16, under the first period, 18, and subtract, leaving a remainder of 2. Bring down the next period, 66, and annex it to the 2, giving 266 for what is called the dividend. Multiply that part of the root which we have already found, namely, 4, by 20 to get the trial divisor. Set this off to the left. Divide 266 by the trial divisor, 80, approximately. This is 3+, so the next figure of our root is probably 3. Write the 3 as the second figure of the answer, and then add it to the trial divisor 80, giving 83, which is the final divisor. Multiply the final divisor, 83, by the figure of the root just found, 3, giving 249. Subtract this from the dividend, 266, leaving 17. If the product of the final divisor and the last figure of the root happened to be larger than the dividend, it means that the figure last found in the root is too large and the next smaller figure should be taken. Bring down the next period, 24, and annex it to the 17, giving a new dividend, 1724. Repeat the preceding process as follows: multiply that part of the root already found (43) by 20, giving 860 for the new trial divisor. Divide the dividend (1724) by the trial divisor, obtaining 2+ as the next figure of the answer. Write the 2 as the third figure of the answer and also add it to the trial divisor, 860, giving a final divisor of 862. Multiply this by 2 and subtract from 1724. There is no remainder, and as there are no more periods in the number, the root is complete. Therefore, 186624 432, Answer. = 70. Square Roots of Mixed Numbers.-If it is required to find the square root of a number composed of a whole number and a decimal, begin at the decimal point and point off periods to the right and left. Then find the root as before. Explanation: Taking the nearest square root of the first period, 2, we get 1 for the first figure of the root. Square the 1, subtract from the first period, and bring down the next period, giving a dividend of 157. The trial divisor is then 20, which goes into 157 about 7 times. If we try 7 as the next figure of the root, the final divisor will be 27, which multiplied by 7 makes 189. This is larger than 186, the dividend; therefore, 7 is too large and we must use 6 instead of 7 for the next figure of the root. Since 57 was the last period preceding the decimal point, we then place the decimal point after the 6 in the root. The final divisor is 26. Multiply by 6 and subtract from 157, leaving a remainder of 1. Bring down the next period, 86, making a new dividend 186. The new trial divisor is 320, which we find is larger than the dividend, 186. The next figure of the root is, therefore, 0, so we place a cipher after the 6. In this case, then, simply bring down the next period, 23, and annex it to the previous dividend, making a new dividend of 18,623. We also find a new trial divisor 3200, and proceed anew. The next figure of the root is 5. After multiplying and subtracting from the dividend there is still a remainder, 2598. In case a root does not come out exactly, we can continue the process by adding additional ciphers to the number. The next period will then be two zeros and the next figure of the root turns out to be 8. There is still a remainder but we have the root to three decimal places which is quite accurate enough for ordinary work. We place a plus sign after the 8 to indicate that there are more places which have not been calculated. 71. Square Roots of Decimals. Sometimes in the case of a decimal, one or more periods are composed entirely of ciphers. The root will then contain one cipher following the decimal point for each full period of ciphers in the number. |