The tens below by units above, give... 150 Product or square, Nów observe, that this square is made up of the square of the tens, twice the product of the tens into the units, and the square of the units. We must now suppose ourselves ignorant of the real root, 35, and by the above principle, find it. First, divide the power off into periods of two figures each (Art. 126), and it will stand thus: 12'15. Being two periods, there will be two figures in the root, a unit and a ten. Now, the square of the tens can form no part of the unit period, as the square of one 10 produces 100, and the square of tens generally produce hundreds, and nothing less. Hence, the greatest square contained in 12, the 2d period, is the square of the tens; but 9 is the greatest square number less than 12, and 3 its root; and so far, the operation may stand thus: 12'25(3 9 325 But the tens and units are twice multiplied together, and their product forms no part of the units in the power; hence, if we divide the 32 tens, by double the tens in the root, we must have the other factor, or the units of the root. Now the operation stands thus : 12'25(35 (ART. 128.) Let us now illustrate the mode of extraction, by means of a geometrical square. We cannot demand the square root of any number, without first supposing it to represent the area of some square surface, and the root is the length of a side of the supposed square; but to bring it before the mind in a definite manner, we demand the square root of 1849. This number, then, must represent a square. Suppose the figure E B A ABCD contains 1849 square feet, and that the G number consists of two Now double GH, or HI, which is 40, for a divisor, omitting the cipher to leave place for the next quotient figure, to complete the divisor. 80 into 249 are contained 3 times; this 3 is the width of the oblong AEHG, or HFCI. But the square is imperfect without EBFH; then annex the three to the divisor. Now multiply this perfect divisor by the last figure of the root, to get the quantity in the two oblong figures, and the small square which comprises the great square ABCD. From either of the preceding modes of examining this subject, we draw the following RULE. 1. Separate the given number into periods, of two figures each, beginning at the unit's place. 2. Find the greatest square contained in the left-hand period, and set its root on the right of the given number; subtract said square from the left-hand period, and to the remainder bring down the next period for a dividual. 3. Double the root for a divisor, and try how often this divisor (with the figure used in the trial thereto annexed) is contained in the dividual: set the number of times in the root; then, multiply and subtract as in division, and bring down the next period to the remainder for a new dividual. 4. Double the ascertained root for a new divisor, and proceed as before, till all the periods are brought down. [NOTE. When all the periods are brought down, if there be a remainder, annex ciphers to it for decimals, and proceed till the root is obtained to a sufficient degree of exactness. Observe, that the decimal periods are to be pointed off from the decimal point toward the right hand; and that there must be as many whole number figures in the root, as there are periods of whole numbers, and as many decimal figures as there are periods of decimals.] EXAMPLES. 1. What is the square root of 729? Ans. 27. Ans. 327. Ans. 655. Ans. 667. Ans. 881. Ans. 997. Ans. 5464. 8. What is the square root of 141225,64 ? Ans. 375,8. N. B. When the power is a decimal, commence to point off periods from the decimal point, and operate as in whole numbers. EXAMPLES. 1. What is the square root of ,00008836? 2. Find the square root of ,00529. 3. Find the square root of ,002916. Ans. ,0094. Ans. ,023. Ans. ,054. Observe, the roots are greater than the powers, as they should be in fractions. (ART. 129.) Mixed numbers may be square, when reduced to the fractional form (improper fractions); but the square form will commonly be indicated by the denominator of the fraction being a square number. In such cases, reduce the mixed numbers to improper fractions, and extract the roots of numerator and denominator. Ans. 64. EXAMPLES. 1. What is the square root of 3738 ? 2. What is the square root of 2721 ? Ans. 16. Fractions may be square in value, but not expressed by square numbers; but can be reduced to such, and af terwards the square root extracted. amples will illustrate : 6. What is the square root of 230 49 147 ? 7. What is the square root of 27? 8. Find the square root of 22. 9216 The following ex 9. What is the square root of 7956? 10. What is the square root of 2704? 4225 When fractions or mixed numbers are surds, reduce the fractions to decimals, and proceed as in whole numbers. EXAMPLES. 1. What is the square root of 7? Ans. 2,7961+ 2. Find the square root of 173. 3. Find the square root of 2. 3 11 4. Find the square root of APPLICATION. Ans. 4,168+ Ans. ,86602 + Ans. ,9574+ Ans. 9,27+ 1. A certain square pavement contains 20736 square stones, all of the same size; what number is contained in one of its sides? Ans. 144. 2. If 484 trees be planted at an equal distance from each other, so as to form a square orchard, how many will be in a row each way? Ans. 22. 3. A certain number of men gave 30 shillings 1 pence for a charitable purpose; each man gave as many pence as there were men: how many men were there? Ans. 19. (ART. 130.) To be scientific and skillful in the application of square root, it is necessary to attend to the following properties of square numbers. I. A square number, multiplied by a square number, the product will be a square number. II. A square number, divided by a square number, the quotient is a square. III. If the square root of a number is a composite |