From these examples and illustrations, we see that the square of the sum of any two numbers is equal to the square of the first, plus twice the product of the first into the second, plus the square of the second. 5. Find by this method the square of 4+3; 5+8; 1+1; 21=2+4; 54; 6; 124; 20; 30. 143. Let us now extract the square root of 1444. 8 1. As the square contains 4 figures, the root must contain 2 figures, tens and units; and the whole square must consist of the square of the tens, and twice the product of the tens into the units, and the square of the units. 2. As the square of tens is hundreds, the square of the tens of the root will be found in the hundreds of the square. The greatest square in 14 is 9; and its root, 3, is the tens of the root. Subtracting the square of the tens from the whole square, the remainder contains twice the tens of the root into the units, and the square of the units; or the product of twice the tens, plus the units, multiplied by the units. (Why?) (97.) Therefore, dividing the remainder by the former factor, viz., twice the tens plus the units, would exactly give the latter factor, viz., the units. (Why?) But, as the units' figure is not yet found, we take twice the tens, or 6, as a trial divisor. Dividing the 54 tens by this trial divisor, gives 9 for the units of the root. Adding this to the trial divisor for a complete divisor, and multiplying the latter by the units, we get too large a product. Consequently, our units' figure was too large, as was to have been expected, since our trial divisor was too small. We, therefore, erase this units' figure and the product; and, trying a smaller units' figure, proceed as before. Hence the following RULE FOR EXTRACTING THE SQUARE ROOT. 1. Separate the given number into periods of two figures each, by placing a dot over every second figure, beginning at units; thus, 84165.041680. The number of dots will show how many figures, whether integers or decimals, the root will consist of. 2. Find by trial the greatest square number in the left hand period, (which may consist of one or two figures,) and place its root on the right hand of the given number, as you do a quotient in division. Subtract the square of the root thus found from the first period, and to the remainder bring down the second period for a dividend. 3. Double the root already found, and place it on the left of this dividend, for a trial divisor. See how many times it is contained in the tens of the dividend, and annex the result both to the root already found, and also to the trial divisor, for a complete divisor. If the trial divisor is not contained in the tens of the dividend, annex a naught to the root, and to the trial divisor, for the next trial divisor, and bring down the next period for a dividend. 4. Multiply the complete divisor by the last figure in the root, subtract the product from the dividend, and to the remainder annex the next period for a new dividend. If the product should exceed the dividend, diminish the last figure of the root, and of the complete divisor. 5. Repeat the same process, viz.: Double all the figures in the root for a new trial divisor, and, dividing by it, find as before the next figure in the root; and continue the operation till all the periods are brought down. To extract the root of a common fraction, reduce it to its lowest terms, and extract the root of the numerator and of the denominator, for the numerator and denominator of the root, when both terms of the fraction are perfect squares; if they are not, reduce the fraction to a decimal, and extract the root by the above rule. If the number of decimals is odd, a naught must be annexed; and if there is a remainder after bringing down all the periods, decimal naughts forming new periods may be annexed. PROOF. Square the root; the result should equal the given square. 1. Extract the square root of 786897. As the decimal .875 consists of an odd number of figures, a naught is annexed to make the number even. The second trial divisor, 56, not being contained in the tens of the dividend, a naught is annexed to the root and to the trial divisor, and another period brought down, as directed in the rule. 78689.8750(280.517 4 48) 38,6 5605) 2898,7 561027) 401490,0 87711 2. Extract the square root of 1225; 2401; 7569. 3. Extract the square root of 15625; 531441; 1048576. 4. Extract the square root of 20421361; of 36529936; of 72726784. 5. Find 12.25; 24.01; .7569. 6. Find 5314.41; 53.1441; 1.5625; 104.8576. 7. Extract the square root of 2042.1361; of 204.21361. 8. Extract the square root of; of 324; of 4288. 9. What is the square root of 12? (Change it to an improper fraction.) Of 4561? of 1411? 10. What is 3μ? √253? √24.301? /45164.8106? NOTE. Carry the root in Nos. 10 to 15, to six decimal places. 11. What is the square root of 30.4167081? of 31⁄2? of 5043 ? 12. What is the square root of 410.680716? of .00081645? of by? 35 13. Extract the square root of Ts; of; of 8.7165 14. Find NOTE. Add the quantities before extracting the root. 8 15. Find 35+ of 275; No33 ÷ 15. For the application of the square root, see Art. 160 to 165, which the pupil may learn before proceeding to the cube root, if his teacher should think it best. QUESTIONS. What is involution? a power? the first power? the root? Why is it called the root? What is the second power? the third? What is the square of a number? Why so called? The cube of a number? Why? How is a power indicated? Give examples. Repeat the rule for finding any required power of a number. What is evolution? What is the difference between involution and evolution? What is the root of a number? the square root? the cube root? How is the square root indicated? the cube root? the 4th root? In what other way? Give examples. What is a perfect power? a rational number? an imperfect power? a surd? Show that a number may be a perfect power of one degree, and an imperfect power of another. What is the extraction of the square root? Repeat the first 12 integral numbers, and their squares. How many figures does a square number contain? how many decimals? Prove it to be true. How may the number of figures in the root be known? To what is the square of the sum of two numbers equal? Illustrate this. Repeat the rule for extracting the square root, and show the reason for the different parts of the process. How do you extract the root of a common fraction? When is it to be reduced to a decimal? 144. EXTRACTION OF THE CUBE ROOT. The Extraction of the Cube Root of a number is the process of finding one of its three equal factors; or, of finding a number which, being multiplied into itself, and then into that product, will produce the given number. The following numbers in the upper line represent roots, and those in the lower line. their third powers, or cubes. Roots. 1 2 3 4 5 6 7 8 9 10 12. Cubes. 1 8 27 64 125 216 343 512 729 1000 1331 1728. The cube of any number cannot have more than three times its number of figures, and never but two less than three times as many. This is true when the root has but one figure; for 13 = 1; 23=8; 3=27; 53= 125; 9=729. It is also true when the root has two figures; for 10a= 1000; 503=12500; 993=970299. Show that it is true when the root consists of three figures; four figures. There will be three times as many decimals in the cube as in the root. 1. What is the cube of .1 .03? .005? .0007? Hence the number of figures in the root, both of integers and decimals, may be known by placing a dot over every third figure in the cube, beginning at units. 2. How many figures will there be in the root, if there are 11 in the cube! 12 13 14 15 20+8X2048. Multiply ( 202 2×20X8+8= 12048 203+2×20a×8+20×82=(202+2X20X8+82)X20 209×8+2×20×82+8=(202+2×20×8+82)X8 20843X202X8+3X20X82+83=(20+8)X(20+8)X(20+8). By cubing 28, as in the above example, we see that the cube of the sum of two numbers, consisting of units and tens, is the cube of the tens,+3 times the square of the tens multiplied by the units, +3 times the tens multiplied by the square of the units, the cube of the units; or, The cube of the tens; plus the product of 3 times the square of the tens, increased by 3 times the tens multiplied by the units, and the square of the units, all multiplied by the units (97); plus the cube of the units; that is, 203+ (3 × 202 × 8) + (3 X 20 X 82)+83=203+ (3 × 202+3X 20 X 8+82) × 8+83. 145. We will now extract the cube root of 21952. 1. The root will contain two figures, 8 tens and units. (Why?) 2. The cube t. d. 1200 21952(29 540 8 13952 16389 13952 480 13952 64 c. d. 1821 of the tens will be found in the thousands. the 'square of the tens, &c., would exactly give the latter factor, viz., units. But as the units' figure is not yet found, we will take three times the square of the tens, or 12 hundreds, as a trial divisor. If we divide the remainder by this trial divisor, the quotient would be more than 11. We know the units' figure cannot be more than 9; therefore the quotient is too large, as we might expect it would be, since our divisor is but a partial divisor. A complete divisor is to be formed by adding to the trial divisor 3 times the tens multiplied by the units, and the square of the units. 3 times 2 tens multiplied by 9, or, (which is the same,) 30 times 2 multiplied by 9, is 540, and 92 is 81. Adding these to the trial divisor for a complete divisor, and multiplying by 9, we get too large a product. Our quotient figure is therefore still too large. We must erase it, and go over this. part of the work again. Taking 8 as a quotient figure, we make the complete divisor 1744, which being multiplied by 8, gives the product 13952. Hence the RULE FOR EXTRACTING THE CUBE ROOT. 1. Separate the given number into periods of three figures each, by placing a dot over every third figure, beginning at units; thus, 31486.100840. (What will the dots show? Why?) 2. Find by trial the greatest cube number in the left hand period, (which may consist of one, two, or three figures,) and place its root on the right. Subtract the cube of this root from the first period, and to the remainder bring down the 2d period for a dividend. 3. Take 300 times the square of the root already found for a trial divisor. Divide the dividend by this trial divisor, and place the result in the root. (If the trial divisor is not contained in the dividend, annex |