POWERS AND ROOTS. 295. A product obtained by using a number twice as a factor is called the second power or the square of the number; thus, 25, (5 x 5), is the second power, or the square of 5. NOTE.—Twenty-five is called the second power of 5, because it may be obtained by using 5, twice as a factor. It is called the square of 5, because it is the number of square units in a square whose side is 5 linear units. 1. What is the second power of 2 ? 8*? 3? 5 ? 2. What is the square of 4? 7 ? 1 ? 6? 9? 10 ? 112 ? 122 = ? 132 E? 142 ? 152 = ? 16' = ? 172 = ? 182 = ? (a) Find the sum of the eighteen squares. 296. The square root of a number is one of the two equal factors of the number. The radical sign, V , (without a figure above it) indicates that the square root of the number following it, is to be taken; thus V 64, means the square root of 64. 1. What is the square root of 144 ? 81 ? 49 ? 2. What is the square root of 36 ? 25 ? 16? V9 = ? V 64 = ? V 121 = ? V100 = ? V4 = ? V1 = ? 400 = ? V169 = ? (b) Find the sum of the fourteen results. Powers and Roots. 297. Any number that can be resolved into two equal factors is a perfect square. 1. Tell which of the following are perfect squares and which are not: 9, 10, 12, 16, 18, 25, 32, 36. NOTE.—It is a curious fact that no number, either integral or mixed, can be found which, when multiplied by itself, will give as a product 10, or 12, or 14, or any number that is not a perfect square. 2. Any integral number that is a perfect square is composed of an even number of like prime factors; that is, its prime factors are an even number of 2's, 3's, 5's, 7's, etc. 3. Tell which of the following are perfect squares : 144, (2 X 2 X 2 X 2 X 3 X 3); 250, (2 x 5 x 5 x 5); 225, (5 X 5 X 3 X 3). RULE.—To find the square root of an integral number, that is a perfect square, resolve the number into its prime factors and take half of them as factors of the root ; that is, one half as many 2's, 3's, or 5's, etc., as there are 2's, 3's, or 5's, etc., in the factors of the number. 4. Find the square root of 1225. 1225 = 5 x 5 x 7 x 7. 71225 = 5 x 7 = 35. 5. Find the square root of 441; of 400. 6. Find the square root of 576; of 324. 7. Find the square root of 784; of 2025. 8. Find the square root of 625; of 3025. (a) Find the sum of the last eight results. Powers and Roots. 298. THE SQUARE OF COMMON FRACTIONS. 1. The square of }, (1 x 1), is Note.—A square whose side is } (of a linear unit) has an area of 1 (of a square unit). Show this by diagram. 2. Answer the following and illustrate by diagram if necessary : () = ? (1) = ? (3)2 = ? ()’ = ? (3)' = ? () = ? (3) = ? (a) Find the sum of the eight results. 3. A square of sheet brass whose edge is of a foot is what part of a square foot ? 299. THE SQUARE ROOT OF COMMON FRACTIONS. 1. The square root of is Note 1.—A square whose area is 1% (of a square unit) is 1 (of a linear unit) in length. Show this by diagram. NOTE 2.—Only those fractions are perfect squares which, when in their lowest terms, have perfect squares for numerators and perfect squares for denominators. 2. What is the square root of ? Of 28? V 144 = ? ? V 96= ? V 141 = ? (b) Find the sum of the seven results. Of į? 25 1 69 3. The area of a square piece of sheet brass is 8 of a square foot. What is the length of the side of the square ? 4. How long is the side of a square of zinc the area of which is one of a square yard ? Powers and Roots. 300. THE SQUARE OF DECIMALS. 1. The square of .5 is NOTE.—A square whose side is .5 (of a linear unit) has an area of .25 (of a square unit). Show this by diagram. 2. Answer the following and illustrate by diagram if necessary: .1? ? .22 = ? .32 ? .42 ? .62 = ? .72 ? ? 1.2? = ? 1.52 ? 1.62 ? 1.82 ? (a) Find the sum of the twelve results. 3. A square of sheet brass whose edge is .9 of a foot is what part of a square foot ? = ? = .82 301. THE SQUARE Root OF DECIMALS. 1. The square root of .25 is Note 1.—A square whose area is .25 (of a square unit) is .5 (of a linear unit) in length. Show this by diagram. NOTE 2.—Only those decimals are perfect squares which, when in their lowest decimal terms, have numerators that are perfect squares and denominators that are perfect squares. The decimal denominators that are perfect squares are 100, 10000, 1000000, etc. 2. What is the square root of 40%? Of .36 ? Of .64 ? V 14 4 = ? V1.44 = ? V 2.25 = ? V 6.25 = ? (b) Find the sum of the seven results. 3. How long is the edge of a square of zinc whose area is 4.84 square feet?* 100 *4.84 feet is 183 feet. Powers and Roots. 302. A product obtained by using a number three times as a factor is called the third power, or the cube, of the number; thus, 125 (5 x 5 x 5) is the third power, or the cube, of 5. NOTE.—One hundred twenty-five is called the third power of 5, because it may be obtained by using 5 three times as a factor. It is called the cube of 5 because it is the number of cubic units in a cube whose edge is 5 linear units. 13 = 1 33 = 27 53 = 125 73 = 343 93 = 729 1. Find the cube of 12; of 13; of 14; of 15. 163 = ? 173 ? 183 ? 198 20% = ? (a) Find the sum of the nine results. = ? 303. The cube root of a number is one of its three equal factors. The radical sign with a figure 3 over it indicates that the cube root of the number following it is to be taken; thus, Ŭ 512, means, the cube root of 512. RULE.- To find the cube root of an integral number that is a perfect cube, resolve the number into its prime factors and take one third of them as factors of the root; that is, one third as many 2's, 3's, or 5's, etc., as there are 2's, 3's, or 5's in the factors of the number. 1. Find the cube root of 216. 216 = 2 X 2 X 2 X 3 X 3 X 3. 216 = 2 x 3 2. Find the cube root of 1728; of 3375; of 2744; of 10648; of 5832. (b) Find the sum of the five results. 3 |