6. Find the cube of all the numbers from 1 to 12. 9. Cube,, 1, †, 1, †, 21. 10. Square .4, .7, .9, .03, .05, .12, .08, .07, .06. 11. Cube .2, .04, .30, .1, .10, .20, .5. 12. How often is 3 used as a factor in finding the fifth power of 3? 8 in finding the second power of 8? 13. The cube of 6 equals 6 used how often as a factor? The fifth power of 6? The seventh power? 14. What power of 5 is 58 multiplied by 52? What power of 7 is 74 multiplied by 73 ? 15. What power of 12 is equal to 122 × 128 × 124 ? What power of 4 is equal to 45 × 42 ? 16. What power of a number is the square of its square? The cube of its cube ? 17. What power of 3 is the square of its cube? The cube of its square? 18. What power of 6 is equal to 65 divided by 62? What power of 8, to 84 divided by 88 ? 19. What number multiplied by 14 gives of the square of the number? By 10, gives of its cube? 20. What power of 4 is the cube of the cube of 42 ? 2. What is the 8th power of 3? The 9th power of 2 ? The 10th power of? Of 56? Of 100? Of 26? Of 65? Of 75 ? 6. What is the square of 63.05? Of 10.21 ? 7. What is the square of 61? Of 91 ? Of 121 ? Find the value of each of the following expressions : 1764 = (40 × 40) + 2 (40 × 2) + (2 × 2) SOLUTION.-The number 42 consists of 40 units + 2 units, or 4 tens + 2 units. Multiplying each part separately, the results under a = those under b. Or, by substituting letters, we get the result t2 + 2tu + u2. Hence, The square of a number expressed by two figures, equals the square of the tens, plus twice the product of the tens into the units, plus the of the units. ike manner find the square of each of the following Ders: 1. 24; 35; 52; 81; 93; 19; 64; 75; 27; 83; 72; 91. 15; 26; 39; 47; 86; 58; 95; 66; 53; 38; 65; 49, 258 = 208+3 (202 × 5) +3 (20 × 52) +53 = 15625 Hence, the cube of a number expressed by two figures, equals the cube of the tens, plus three times the product of the square of the tens into the units, plus three times the product of the tens into the square of the units, plus the cube of the units. In like manner find the cube of each of the following numbers: 1. 32; 16; 24; 31; 44; 56; 72; 82; 54; 39; 26. 2. 15; 29; 37; 68; 94; 59; 45; 63; 54; 35; 47. When the number consists of three or more figures, we complete the square or cube, according to the following formulas: (h+t+u)2= h2+2ht+t2+2 (h+t)u+u2. 4. 870. Ans. 756900. 8. 40.95. Ans. 1676.9025. Raise to the third power: 9. 100. 10.999. Ans. 1000000.|13. 1000. Ans. 1000000000. Ans. 997002999. 14. 2.002. Ans.8.024024008. 11. 6.05. Ans. 221.445125. 15. 1.001. Ans.1.003003001. 12. .004. Ans. .000000064.|16. 4.4. 17. .0152. 18. 2.612. 19. 2103. Ans. 85.184. Ans. .1111.. Ans. .000225.120. (8.251)2. Ans, 68.14501. EVOLUTION. 184. Evolution is the process of finding the root of a number. 185. The Root of a number is one of the equal factors that will produce it. Thus, 4 is the root, or one of the two equal factors which will produce the number 16. 186. The symbol, √, called the Radical Sign, when placed before a number, indicates that its root is to be taken. To denote the degree of the root, a small figure, called the index, is written in the angle of the symbol. Thus, 8 indicates the cube root of 8. NOTE. The symbol without any index, always denotes the square root. 187. Roots may also be expressed by fractional exponents; as 61 6, and signifies the cube root of the first = power of 6; 63 = 62, and signifies the cube root of the second power of 6. The numerator denotes the power to which the quantity is to be raised; and the denominator, the root that is then to be taken. 188. As any number whatever may be considered as a power whose root is to be extracted, it follows that there are two kinds of roots, the Rational and the Surd. 189. A Rational or exact root, is the root of a perfect power; as, 2 is the rational root of 4. 190. A Surd is the indicated root of an imperfect power; as, √5 denotes the square root of the imperfect power 5. 191. The roots of numbers are named from their corresponding powers, as follows: The square root of 4 is 2, expressed √4 2. The cube root of 8 is 2, expressed = 82. The fourth root of 81 is 3, expressed 81 = 3. The fifth root of 32 is 2, expressed /32 = 2. SQUARE ROOT. 192. The Square Root of a number is one of its two equal factors; as, 9 is the square root of 81. To assist the pupil in finding the number of places in the square root of any number, we give the following TABLE. 12= 9281 1 Period. 102 = 1,00 2 Periods. PRINCIPLES. I. The square of any number contains twice as many figures, or twice as many less one, as the number itself. |