SQUARE ROOT OF NUMBERS. 109. The SQUARE ROOT of a number is one of its two equal factors. Thus, 25 = 5 x 5; hence, 5 is the square root of 25; that is, √25 = 5, or (25)✈ = 5. The following table, verified by actual multiplication, is employed in finding the square root of any number less than 100. TABLE. 1 2 1 4 9 16 25 36 49 64 81 100. Powers. 3 4 5 6 7 8 9 10. Roots. To employ the table for finding the square root of a number less than 100. Look for the number in the first line; if it is found there, its square root will be found immediately under it; if it is not found there, it will fall between two numbers in that line, and its square root will be found between the two numbers immediately below; the lesser of the two will be the entire part of the root, and will be the true root to within less than 1. If a number is greater than 100, it square root will be greater than 10, that is, it will contain tens and units. Let N denote such a number, x the tens of its square root, and y the units; then will N = (x + y)2 = x2 + 2xy + y2 = x2 + (2x + y)y. That is, the number is equal to the square of the tens in its root, plus twice the product of the tens by the units, plus the square of the units. Since the square We first find the tens of the root. of tens can contain no significant figure less than hun dreds, the two figures on the right may be pointed off То and the square of the tens will be found in the number to the left of the point. Having found the tens, we subtract their square from the given number, and the remainder will be made up of twice the product of the tens by the units plus the square of the units. find the units, divide the remainder by twice the tens, and the quotient will be the units, or a number greater than the units. To test it, add it to twice the tens, and multiply the sum by the number found; if the product is equal to the remainder, or less than it, the num ber found is the root sought; but if greater, diminish it by 1, and test as before, till the correct number is found. In dividing the remainder by twice the tens, the last figure may be omitted, since twice the product of the tens by the units can never give a significant figure in the units' place. EXAMPLE. Find the square root of 1764. Pointing off the two right hand figures, there remains the number 17, the greatest perfect square in which is 16. The square root of 16, or 4, is therefore the number of tens of the required root. OPERATION. 176442 16 164 Write this on the right, after the manner of a quo tient. Subtracting 16 hundreds (the square of 4 tens) from the number, the remainder is 164. Doubling the tens, we have 8, which is contained in 164, exclusive of the right hand figure, 2 times, which is the number of units adding this to the 8 tens (which is done by writing it after 8), and multiplying the sum by 2, we have 164, equal to the remainder. Hence, if we write the 2 after the 4, in the root, we shall have the roo sought equal to 42. When the number contains more than 4 places of figures, we point off two figures from the right, as before. The operation is then reduced to finding the square root of the remaining numbers, that is, the tens of the root. In finding this root, for the same reason as before, we point off another period of two figures, and the operation then is reduced to finding the square root of the remaining number, that is, the tens of the tens, or hundreds of the root. If this number containg more than two places of figures, for the same reason as before, we point off another period of two figures to the right, when the square root of the remaining number will give the tens of hundreds, or thousands of the root; and so we continue pointing off. till a number is found on the left containing less than three places of figures: the operation is then but a successive applica tion of that already explained. Hence, the RULE. I. Point the number off into periods of two figures cach, beginning at the units' place. II. Find the greatest perfect square in the first period on the left, and place its square root on the right, after the manner of a quotient in a division; then subtract the square of this number from the first period, and bring down the next period for a remainder. III. Double the root already found, and see how often it is contained in this remainder, exclusive of the right hand figure; write this quotient for a second figure of the root, annex it also to the divisor used: multiply. the divisor thus increased by the quotient figure found, subtract this product from the first remainder, and bring down the next period for a second remainder. IV. Double the root already found and proceed as before, continuing the operation till every period has been employed. If the final remainder is 0, the root is exact, if it is not 0, the root found is true to with in less than 1. SQUARE ROOT OF FRACTIONS. 110. It has been shown (Art. 100) that a fraction is squared by squaring both numerator and denominator separately reversing the process, we have for extracung the square root of a fraction the following RULE. Extract the square root of the numerator for a new numerator, and the square root of the denominator for a new denominator. 111. When the terms of the fraction, after being reduced to its simplest form, are not perfect squares, the exact root is then .impossible. In such cases, we inay multiply both terms by any number which will make the denominator a perfect square. Then, extracting the square root of the numerator to the nearest unit for a new numerator, and the square root of the denominator for a new denominator, the resulting fraction will be the true root to within less than the fractional unit. |