ᏢᎪᎡᎢ Ꮩ POWERS AND ROOTS 295. 3 x 3 = 9. 3 is used twice as a factor to give 9. 9 is called the second power of 3. What number is the second power of 2? of 4? of 5? of 10? of 1? of 12? 2. The second power of a number is called its square, as the number of units in the area of a square surface is found by taking the second power of the number denoting the length of a side of the square. 3. The square of 3 may be indicated thus: 32. Indicate the square of 4; of 5; of 1; of 10; of 12. Give the value of each: 72, 82, 22, 62. The small figure written at the right and above indicates how many times the number is to be taken as a factor and is called the exponent of the number. 4. 3 × 3 × 3 = 27. 27 is the third power, or cube, of 3. What number is the cube of 1? of 2? of 4? of 5? of 6? of 10? of 12? 5. The cube of 3 may be indicated by an exponent, thus 33. Indicate the cube of 7; of 8; of 9. Give the value of each: 13, 23, 103, 128. 54 is read the fourth power of 5, or 5 to the fourth power; it means 5 × 5 × 5 × 5. Read and tell meaning of: 64, 35, 26. 6. Find the volume of a cube whose edge is 5 in. Find the cube of 5. 7. Give the square of each of the numbers from 1 to 12. 8. Square 1, 1, 4, .5, 1.5, .04, 16, 21. 9. Find and memorize the cubes of 1, 2, 3, 4, 5, 6, 10, 12. The process of finding a power of a number is sometimes called involution. 10. A number that is the square of some integer or fraction is called a perfect square. Thus, 25 (5 × 5) and 25 (× 5) are perfect squares. Is 24 a perfect square? 11. Square each: 20, 30, 40, 50, 60, 70, 100. 12. Is the square of 2 plus the square of 3 the same as the square of 5? 296. 1. Which is the more and how much, 202 + 52 or 252 ? . 2. The square of any number composed of tens and units may be found thus: 400+2(100) + 25 = 202+ 2(20 × 5) +52 tens and the units, plus the square of the units. 3. Square as above: 23, 47, 105 (100 + 5). 4. The figure represents a square whose side is 25 units. The square whose side is 20 units contains 400 square units. The two rectangles 20 by 5 contain 100 square units each. The square is completed by the addition of the small square 5 by 5, containing 25 square units. The area of the square is (400 + 2(20 × 5)+ 25), or 625 square units. 5 20 5 100 25 400 1000 20 5 5. Construct a square whose side is 10 + 5 units. 20 297. Roots. 1. Since 9 is the square of 3, 3 is the square root of 9; that is, it is one of the two, equal factors of 9. What number is the square root of 4? of 25? of 64? of 36? of 49? of 16? of 144? of 100? of 81? of 121? of 1? 2. Since 27 is the cube of 3, 3 is the cube root of 27. What is meant by the cube root of a number? What number is the cube root of 1? of 125? of 8? of 1000? of 1728? 3. The sign (√ is called the radical, or root sign, and is placed over a number to show that its root is to be taken. The root to be taken is indicated by a small figure, called an index, written in the radical thus, is read the cube root of 27. The index 2 for is usually omitted. 27, which square root 4. Read and give the roots: √64, 64, √49, √100, $125, √81, √36, √144, 81. The process of finding the root of a number is sometimes called evolution. 298. Finding Roots by Factoring. Roots of perfect squares may be found by factoring. 1. Find the square root of 324. By factoring, 324 = 2 × 2 × 3 × 3 × 3 × 3. Arranging the factors into two like groups, Factor 2774. Group the factors into three like groups. The product of one of these groups is the cube root. 3. The square root of a fraction is the square root of its numerator over the square root of its denominator, thus: √ = 1 300. 1. Compare √1 = 1, √1|00 = 10, and √1|00|00 = 100. Notice that there is one figure in the square root for each period of two figures each into which the square can be separated, beginning at units. The period at the left may contain only one figure. By separating any number into such periods, the number of figures in the square root may be told. 2. How many figures are there in the square root of each 11,664? 129,600? 11,025? 3. 1.22 1.44; 9.92 98.01; 1.222 1.4884. Notice = = = that there are two decimal places in the square for each decimal place in the root. 4. How many decimal places are there in the square root of each: 4.1616? 1190.25? 2550.25? 301. Square Root. a. Find the square root of 529. b. Find the side of a square whose area is 529 square units. As the square root of some numbers cannot be found by factoring, another method of finding the square root of numbers is necessary. From Sec. 296 we see that the square of a number is the square of the tens, plus twice the product of the tens and units, plus the square of the units; and from Sec. 300 we see that the number of fignumber is the same as the ures in the square root of any number of periods of two places each, beginning with units into which the number can be separated. MODEL: 202 = 4 2 x 2040 129 (40+3) × 3 =129 a. As 529 can be separated into two 5'29 23 periods, its square root consists of tens and units. Since the square of tens is hundreds, 5 hundreds must include the square of the tens of the root. The largest perfect square in 5 hundreds is 4 hundreds. The square root of 4 hundreds is 2 tens. Write this in the answer at the right. The square of 2 tens is 4 hundreds. Subtract 4 hundreds from 529. The remainder is 129. This remainder must be twice the product of the tens and the units, plus the square of the units. Twice 2 tens is 40. The units' figure of the root is found by taking 40 as a partial divisor. 40 is contained in 120 (omitting the 9, as it is evidently the square of the figure in units' place, or a part of its square) three times. Write 3 as the units' figure of the root. Use 43 as the complete divisor. 3 x 43 = 129, which exhausts the remainder. 3 20 B A 20 400 09 20 C b. As the largest perfect square in D3 5 hundred square units contains 4 93 hundred square units, its side is 20 units (4). 129 square units remain to be added in such form as to keep the figure a square. It is evident that these units must be added along two adjacent sides, as B and C, and at the corner, as D. The combined length of the two rectangles, B and C, is 40 units. Their width may be determined from the fact that their combined areas, plus the area of the small Omitting the 9, as it evidently is the number (or a part of the number) of square units in the small square, 120 square units 40 square units 3, the number of units in the width of the rectangles, and also in the side of the small 20 3 square D, is 129 square units. square. = |