.12 .01. 8. In like manner write the third power of 1, , , .. 9. Write the second power of .1, -2, -3, .4, .5, 6, .7,.8,.9, in this manner : .1 X .1 10. Find the value of .1,.23, .33, .43, -53, .69, .73, -83, .93. 11. Find the value of (), (.4), (3), (4), (.1)*, (.02)". In the equation x = 4%, the value of x is the square of 4. . Process. 16. 20 x = 42 = 4 X 4 16. In like manner find the value of x in the following: 1. X = 25. 8. x = 103. 15. x = .093. .054. .0053. 2.052. (25})? 20. X (4.5004) .1253. (21.653)? .043. INVOLUTION BY ANALYSIS. 32 = NOTE.—If this subject be considered too difficult, it may be omitted. 322 = (30 + 2) = = (tens + units) = (t + u)?. t We will now square 32 as tens and units. 30 + 2 ttu 32 = 30 + 2 ttu 4 ť 1024 ť + 2t xu + up. Hence we have a very important principle : u? { 21 x u = 900 2t X u = The square of any number consisting of tens and units the tens? + 2 times the tens x the units + the units?. To illustrate, find the square of 35 in accordance with this principle. 900 = ť 300 352 = (tens + units)? 25 ua 1225 = t + 2 x 4 + . + x ” In like manner find the square of 45, 56, 97, 21, 38, 63, 75, 88, 19, 24. We will now proceed to find the cube of 32. 323 = 322 x 321. 8 u 120 4 120 3t X u2 60 120 322 x 32 = 60 1800 900 1800 1800 27000 ť 32,768 = t + 3t xu + 3txu + u'. ť Hence the principle: The cube of any number consisting of tens and units the teng3 + 3 times the tens? x the units + 3 times the tens x the units? + the units. For example: 353 = (30 + 5) = 30% + 3 X 30 X 5+ 3 x 3 X 30 X 52 + 53 = 27,000 + 13,500 + 2250 + 125 = 42,875. In like manner find the cube of 35, 22, 19, 53, 38, 27, 47, 48, 45, 63, 73, 77. Involution, as we have seen, develops the power from the root. The reverse process, that extracts the root from the power, is called Evolution. 3f? Хи EVOLUTION. INDUCTIVE STEPS. 1. The numbers represented by the digits and their squares are: Numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9. . Squares, 1, 4, 9, 16, 25, 36, 49, 64, 81. Note.—Pupils should memorize this table of squares and roots. 2. The numbers are the square roots of their squares : 1 is the square root of 1; 2 is the square root of 4; 3 is the square ; : root of 9, and so on. 3. A square number is the product of two equal factors, either of which is the square root of that square number. 64 = 8 X 8; 64 is a square number, and 8 is its square root. 4. Since 27 = 3 X 3 X 3, 27 is a cube number, and 3 is , its cube root; and since 16 2 X 2 X 2 X 2, 16 is a fourth power, and has 2 for its fourth root. 5. Evolution is the process of finding the roots of numbers. A number having an exact root is a perfect power. 6. The radical or root sign is The 64 may be thus expressed : V 64, or thus : 64%. square root of SQUARE ROOT. By Factoring. The prime factors of 16 are 2 x 2 X 2 X 2 = 4 x 4; X therefore V 16 = 4. 36 = 2 X 2 X 3 X3= 6 X 6; thereV : 2 3 fore V 36 6. = In like manner find the square root of: 1. 144. 5. 576. 9. 1764. 13. 3969. Periods and Roots Compared. Separating the following squares into two-figure periods as far as possible, we have: One period. 11' = 1. V81' = 9. Two periods. Three periods. Obviously, one period in the square gives but one figure in the root; two periods in the square, two figures in the root; three periods in the square, three figures in the root. Hence the principle : The number of figures in the square root equals the number of two-figure periods into which the square can be pointed off, beginning at units. NOTE.—The period on the extreme left may contain but a single figure. EXERCISES. 1. In accordance with the foregoing principle, state how many periods the squares of the following roots contain: 4, 9, 32, 99, 317, 999, 3163, 9999, 21, 115, 4156, 19, 316, 6184, 35, 584, 8196. 2. Show that the squares of the numbers from 4 to 9, inclusive, give full periods. Suggestion : 42 16; 92 = 81. 3. Show that the squares of the numbers from 32 to 99, inclusive, give two full periods. 4. Show that the squares of the numbers from 317 to 999, inclusive, give three full periods. 5. Show that the squares of the numbers from 3163 to 9999, inclusive, give four full periods. 6. State the exact number of figures contained in the squares of the following numbers : 3, 5, 21, 29, 41, 66, 97, 125, 200. 7. Point off the following numbers into periods, and tell the number of figures in the root of each : 196, 1296, 2809, 5625, 400,689, 516,961, 182,329, 23,804,641, 9,991,921. Extraction of Square Root. General Method. Find the square root of 190,969. Square. Root. 19'09'69 ( 437 16 309 1. The number pointed off is . 2. The greatest square in 19 is 3 The square root of 16 is 4 (the first figure of the root) 4. The remainder, with the second period, is . 5. Twice the root-figure 4 is 8, the trial divisor. 6. 30 = 8 gives 3 for the second root-figure. 7. 3 annexed to 8 gives 83, the complete divisor. 8. 83 x 3 9. The remainder, with the third period annexed, is 10. Twice 43 or 86 is the second trial divisor. 11. 606 86 gives 7 for the third root-figure. 12. 7 annexed to 86 gives 867, the second complete divisor. 13. 867 x 7., 249 6069 0 In a treatise on arithmetic, a scientific explanation of the square root is scarcely admissible, as methods essentially algebraic or geometrical have to be adopted. |