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. ✓1225 = 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 3), is NOTE.—A square whose side is į (of a linear unit) has an area of $ (of a square unit). Show this by diagram. 2. Answer the following and illustrate by diagram if necessary: (3)= ? (1) = ? (1) = ? ? (6)% = ? 2 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 17 is Note 1.-A square whose area is 16 (of a square unit) is ? (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 foo? Of is? Of į? ✓144 = ? Vid = ? Vit = ? (b) Find the sum of the seven results. 3. The area of a square piece of sheet brass is 184 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 of a square yard ? 5. What is the perimeter of a square piece of land whose area is 16 of a square mile ? 36 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: ? .22 .32 .49 = ? .82 ? 1.22 ? 1.82 ? 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 ? .12 301. 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 400? Of .36 ? Of .64 ? V 144 = ? 1.44 = ? 2.25 = ? 76.25 = ? (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 ? * 4. What is the perimeter of a piece of land whose area is .49 of a square mile ? *4.84 feet is 180 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. Nore.-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. 1. Find the cube of 12; of 13; of 14 ; of 15. 163 = ? 173 = ? 183 ? 193 = ? (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, V 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. 13/ 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. Powers and Roots. 304. MISCELLANEOUS PROBLEMS. 1. Square 42. Then resolve the square of 42 into its prime factors and compare them with the prime factors of 42. 2. Cube 42. Then resolve the cube of 42 into its prime factors and compare them with the prime factors of 42. 3. Square 45. Then resolve the square of 45 into its prime factors and compare them with the prime factors of 45. 4. Cube 45. Then resolve the cube of 45 into its prime factors and compare them with the prime factors of 45. 5. Square 27. Then resolve the square of 27 into its prime factors and compare them with the prime factors of 27. 6. Cube 27. Then resolve the cube of 27 into its prime factors and compare them with the prime factors of 27. 7. Divide the cube of 15 by the square of 15. 8. Divide the cube of į by the square of . 9. Divide the cube of .7 by the square of .7. 10. Divide the cube of 2.5 by the square of 2.5. 11. Find the square root of each of the following perfect squares : 12. Find the cube root of each of the following perfect cubes : |