14'348'907 (243 23 = 8 42 = 16 1456 5824 2402 x 3= 172800 524907 240 X 3 X 3 2160 32 = 9 174969 524907 384. Rule for Anding the Cube Root of a Number. Beginning at the decimal point, separate the given power into periods of three figures each. Find the greatest cube in the left period, and place its root at the right. Subtract the cube of this root from the left period, and to the remainder annex the next period for a dividend. Annex a cipher to the root already found, and take three times its square for a trial divisor. Divide the dividend by this trial divisor, and place the quotient as the next root figure. Multiply the number last squared by the last root figure, and add three times the product and the square of the last root figure to the trial divisor for a complete divisor. Multiply the complete divisor by the last root figure, subtract the product from the dividend, and to the remainder annex a new period. Form a second trial divisor, and proceed as before until all the periods have been used. NOTE. — Note 2, under the rule for square root, applies likewise to cube root. Extract the cube root of the following numbers : 64. 91125. 69. 12977875. 74. What is the cube root of 8.144865728 ? 8.'144'865'728' (2.012 8 120000 144865 600 Here, as 0 occurs in the root, we annex 00 to the trial divi1 sor, 1200, and bring down to 120601 120601 the corresponding dividend 12120300 24264728 another period. 12060 4 75. Find the cube root of 64481.201. 80. What is the value of 0.075686967 ? 81. Find the value of $0.008649 to the nearest thousandth. 82. Find the cube root of 0.000007 to the nearest thousandth. 83. What is the cube root of 25 to the nearest hundredth ? 385. When both terms of a fraction are perfect cubes, the cube root may be found by taking the cube root of each term ; but, if not, reduce the fraction to a decimal, and then find the root. Mixed numbers may be changed either to improper fractions or to mixed decimals. 85. What is the cube root of 4831} ? 86. What is the cube root of 49.394 ? 87. What is the cube root of } ? 88. What is the cube root of jy to the nearest hundredth ? 89. Find the cube root of 81 Á to the nearest hundredth. 90. What is the cube root of 166; ? 91. A bushel contains 2150.42 cubic inches. What is the depth in inches, to the nearest hundredth, of a cubical bin which shall contain 8 bushels ? QUESTIONS. 360. What is a power ? The first power? The second power ? The third power ? 361. The exponent of a power ? 362. What is involution ? 364. What is a root ? The second, or square, root ? The third, or cube, root ? The fourth root ? 365. A perfect power ? 367. What is evolution ? 369. The square of a number contains how many times as many figures as its root ? 378. The cube of a number contains how many times as many figures as its root ? MENSURATION. 386. Mensuration treats of the measurement of lines, surfaces, and solids or volumes. RIGHT-ANGLED TRIANGLES. 387. A Right-angled Triangle has one right angle. The Hypothenuse is the side opposite the right angle, and the Perpendicular is the side perpendicular to the base. A Right-angled Triangle. 388. It will be seen from the diagram that The square of the hypothenuse is equal to the sum of the squares of the other two sides. Hence, 389. To find the hypothenuse, Take the square root of the sum of the squares of the other two sides. 390. To find the base or perpendicular, Take the square root of the difference of the squares of the hypothenuse and the other side. 1. The base of a right-angled triangle is 8, and the perpendicular 6. What is the hypothenuse ? Solution. - 82 + 62 = 64 + 36 = 100; 100 = 10. 2. The hypothenuse is 30, and one of the sides is 18. What is the other side ? Solution. — 302 — 182 = 900 – 324 = 576; V 576 = 24 3. What must be the height of a ladder to reach to the top of a house 20 feet high, the bottom of the ladder being placed 15 feet from the base of the house ? 4. The hypothenuse is 157 feet, and the perpendicular 132. What is the other side ? . 5. Two vessels sail from the same port; one sails due south 48 miles and the other due west 36 miles. What is then their distance from each other? 6. A tree stands upon the edge of a river 100 feet wide, and a line extending from the opposite shore of the river to the top of the tree is 400 feet. What is the height of the tree, to the nearest hundredth of a foot? 7. A park in the form of a rectangle is 40 rods long and 36 rods wide. What is the length in rods of a walk between its opposite corners ? 8. A ladder 32 feet long was so placed in a street as to reach a window 25 feet from the ground, and when it was turned to the other side, without changing the position of its foot, it reached a window 20 feet from the ground. How wide was the street ? QUADRILATERALS. 391. A Quadrilateral is a plane figure bounded by four straight lines. 392. Parallel Lines are lines in the same plane having the same direction. Parallel Lines. is a quadrilateral having its op 393. A Parallelogram posite sides parallel. |