By the square root of a number is meant that number which when squared produces the given number. Thus, 4 is the square root of 16, since 42 = 16. 7 is the square root of 49 for a similar reason. The square root is defined also as one of the two equal factors of a number. Thus, 7 x7 = 49. One of the factors is the square root of 49. The symbol for square root, V, is called the radical sign. It is a degenerate form of the first letter of the word radix, the Latin word for root. The exponent is also used as a sign for the square root. √49, 494 are the two ways of indicating the same process, namely, the extraction of the square root of 49. The number written under the radical sign is called the radicand. Students should fix firmly in mind :— Example 1. What is the square root of 5329? SOLUTION. By trial the square root of 5329 is more than 70 and less than 80. Since 140 is contained in 429 3 times, try 3 as a value of x. (140+3)3429. ..√532970 + 3 = 73. Example 2. V9025 = ? SOLUTION. √9025 = 90+x. ... 8025 = 8100 + 180 x + x2. ... 180 x + x2 = 925. ... (180 + x)x = 925. Since 180 is contained in 925 5 times, try 5 for the next figure of the root. 81 9 25 (180+5)5925. √902590 + 5 = 95. In practice, the work is contracted as follows: Beginning at the decimal point, point off the figures of the number in periods of two figures each. By trial find 9 5 the greatest digit whose square is contained 90.25 in the number denoted by the period to the left. Write it as the first figure of the root, and write also its square. Subtract the latter from the period to the left and bring down the next period. Double the part of the root just found for trial divisor. Find next the number of times the trial divisor is contained in the number denoted by the remainder and the period brought down. Write this result in the quotient and in the divisor and then multiply. 185 9 25 Example. 3 5 12 74 49 9 65 3 74 3 25 49 49 hence SOLUTION. Divide the figures of the number into periods of two as in the previous exercises. Then proceed to extract the square root of the number denoted by the two periods to the left. The answer is obviously 350 + some number. Let x represent this number; Since 700 is contained in 4900 7 times, try 7 as a value of ... √127449 = 3507357. Notice the trial divisor is twice the part of the root found. Therefore in a problem in square root where the radicand is an integer consisting of five or six figures, proceed in exactly the same way as has been done in a problem consisting of four figures. Beginning once more, the solution in its contracted form stands as follows: The trial divisor is always twice the part of the root already found. To extract the square root of a decimal, begin at the decimal point, and proceeding to the right, point off the figures in periods of two; next proceed as if the number were an integer. Thus, in taking the square root of .0225, first point off in periods of two figures each. This gives .02 25. Next extract the root of the number denoted by the figures 225. The result is 15. Hence, the required root is .15. To extract the square root of a number part integer and part decimal, begin at the decimal point, and proceeding to the left, point off the integral part in periods of two figures each; next point off the decimal part in periods of two figures each, beginning at the decimal point. If there are not enough figures in the decimal part to make an exact number of periods, annex a cipher or as many ciphers as are necessary to make the required number of periods. To extract the square root of a fraction when its numerator and denominator are perfect squares is a simple matter. Thus, the square root of 25 is ; the square root of 111, or 81, is g, or 1. To get the square root of a fraction, take the square root of the numerator, and the square root of the denominator, and then write the former result for numerator and the latter result for denominator. The fraction thus found is the required square root. Example 1. What is the square root of 17? SOLUTION. V17 4.123; √36 = 6. |