559. Powers are denoted by a small figure placed above the given number at the tight hand. This figure is called the index or exponent. It shows how many times the given number is employed as a factor to produce the required power. Thus, The index of the first power is 1; but this is commonly omitted; for, (2)'=2. The index of the second power is 2; The index of the third power is 3; The index of the fifth power is 5; &c. That is, 2'=2, the first power of 2; 22=2×2, the square, or second power of 2; Ans. 17, 19. 7. The 2d power of 299. Express the given powers of the following numbers: 9. The 5th power of 228. 560. The process of finding a power of a given number by multiplying it into itselj, is called INVOLUTION. 561. Hence, to involve a number to any required power. Multiply the given number into itself, till it is taken us a factor, as many times as there are units in the index of the power to which the number is to be raised. (Art. 558.) OBS. 1. The number of multiplications in raising a number to any given power, is one less than the index of the required power. Thus, 32-3×3; the 3 is taken twice as a factor, but there is but one multiplication. QUEST-553. How are powers denoted? What is this figure clled? What does it show! What is the index of the first power? Of the second? The third? Fifth ! 560. What is involution? 561. How is a number involved to any required powe: 1 2. A Fraction is raised to a power by multiplying it into itself. Thus, the square of 号 is 号X=ˇ Mixed numbers should be reduced to improper fractions, or the common fraction to a decimal. They may however be involved without reducing them. (Art. 220. Obs.) 3. The process of raising a number to a high power, may often be contracted by multiplying together powers already found. The index of the power thus found, is equal to the sum of the indices of the powers multiplied together. Thus, 2×2-4; and 4×4=2×2×2×2, or 24. So 32X33=3×3×3×3×3, or 35; and 54X53-57. 12. What is the square of 23? Common Operation. 23 23 69 46 529 Ans. Analytic Operation. 23=2 tens or 20+3 units. 23 2 tens or 20+3 units. 60+9 400+ 60 And 400+120+9=529. Ans. It will be seen from this operation that the square of 20+3 contains the square of the first part, viz: 20×20=400, added to twice the product of the two parts, viz: 20×3+20×3=120, added to the square of the last part, viz: 3×3=9. Hence, 562. The square of the sum of two numbers is equal to the square of the first part, added to twice the product of the two parts, and the square of the last part. OBS. 1. The product of any two factors cannot have more figures than both factors, nor but one less than both. For example, take 9, the greatest number which can be expressed by one figure. (Art. 34.) And (9)2, or 9×9=81, has two figures, the same number which both factors have. 99 is the greatest number which can be expressed by two figures; (Art. 34;) and (99)2, or 99× 99-9801, has four figures, the same as both factors have. Again, 1 is the smallest number expressed by one figure, and (1)2, or 1×1 =1, has but one figure less than both factors. 10 is the smallest number which can be expressed by two figures; and (10)2, or 10×10=100, has one figure less than both factors. Hence, QUEST. Obs. How many multiplications are there in raising a number to a given power? How is a fraction involved? A mixed number? 562. What is the square of the sum of two numbers equal to? Obs. How many figures are there in the product of any two factors? How many figures will the square of a number contain? The cube? 2. A square cannot have more figures than double the number of the root or first power, nor but one less. 3. A cube cannot have more figures than triple the number of the root or first power, nor but two less. 4. All powers of 1 are the same, viz: 1; for, 1X1X1X1, &c.=1. 13. What is the square or second power of 123 ? 14. The cube of 135 ? 15. The square of 2880? 16. The 4th power of 10? 17. The 5th power of 5? 18. The 7th power of 6? 19. The 6th power of 7 ? 20. The 8th power of 4? 21. The 9th power of 9? 22. The souare of 2.5? 23. The cube of .012? 24. The square of .00125? EVOLUTION. 00 563. If we resolve 25 into two equal factors, viz: 5 and 5, each of these equal factors is called a root of 25. So if we resolve 27 into three equal factors, viz: 3, 3, and 3, each factor is called a root of 27; if we resolve 16 into four equal factors, viz: 2, 2, 2, and 2, each factor is called a root of 16. And, universally, when a number is resolved into any number of equal factors, each of those factors is said to be a root of that number. Hence, 564. A root of a number is a factor, which, being multiplied into itself a certain number of times, will produce that number. OBS. Roots, as well as powers, are divided into different orders, Thus, when a number is resolved into two equal factors, each of these factors is called the second or square root; when resolved into three equal factors, each of these factors is called the third or cube root, &c. Hence, The name of the root expresses the number of equal factors into which the given number is to be resolved. Roots. 1| 2 3 1 | 4 9 Cubes. 771 81 91 10 | 11 12 144 4 | 5 6 | | 16 25 | 36 8|27|64 | 125 | 216 | 343 | 512 | 729 | 1000 | 1331 | 1728 1 QUEST.-Obs. What are all powers of 1? 564. What is a root of a number? Obs. What does the name of the root express? 565. The process of resolving numbers into equal factors is called EVOLUTION, or the Extraction of Roots. OBS. 1. Evolution is the opposite of involution. (Art. 560.) One is finding a power of a number by multiplying it into itself; the other is finding a root by resolving a number into equal factors. Powers and roots are therefore correla tive terms. If one number is a power of another, the latter is a root of the for mer. Thus, 27 is the cube of 3; and 3 is the cube root of 27. 2. The learner will be careful to observe, that In subtraction, a number is resolved into two parts; 566. Roots are expressed in two ways; one by the radical sign (✔) placed before a number; the other by a fractional index placed above the number on the right hand. 3 denotes the square or 2d root of 4; 27, or 27 4 Thus, √4, or 4' denotes the cube or 3d root of 27; 16, or 16 denotes the 4th root of 16. OBS. 1. The figure placed over the radical sign, denotes the rool, or the number of equal factors into which the given number is to be resolved. The figure for the square root is usually omitted, and simply the radical sign ✔✅ is placed before the given number. Thus the square root of 25 is written √25. 2. When a root is expressed by a fractional index, the denominator, like the figure over the radical sign, denotes the root of the given number. Thus, (25)* denotes the square root of 25; (27) denotes the cube root of 27. 3. A fractional index whose numerator is greater than 1, is sometimes used. In such cases the denominator denotes the root, and the numerator the power of the given number. Thus, 83 denotes the square of the cube root of 8, or the cube root of the square of 8, each of which is 4. 4. The radical sign ✔, is derived from the letter r, the initial of the Latin radix, a root. 1. Express the cube root of 74. 2. The square root of 119. 3. The 4th root of 231. 4. The 9th root of 685. Ans. $74, or 74+ 5. The square root of . 6. The cube root of 4. 7. The 4th root of. 8. Express the 3d power of the 4th root of 6. 9 Express the 2d power of the 3d root of 81. Ans. 63. QUEST.-565. What is evolution? Obs. Of what is it the opposite? Into what are numbers resolved in subtraction? In division? In evolution? 566. How many ways are roots expressed? What are they? Obs. What does the figure over the radical sign denote? What the denominator of the fractional index? 567. A number which can be resolved into equal factors, or whose root can be exactly extracted, is called a perfect power, and its root is called a rational number. Thus, 16, 25, 27, &c., are perfect powers, and their roots 4, 5, 3, are rational numbers. 568. A number, which cannot be resolved into equal factors, or whose root cannot be exactly extracte, is called an imperfict power; and its root is called a Surd, or irrational number. Thus, 15, 17, 45, &c., are imperfect powers, and their roots 3.8+; 4.1+; 6.74, &c., are surds, for their roots cannot be exactly extracted. OBS. A number may be a perfect power of one degree and an imperfect power of another degree. Thus, 16 is a perfect power of the second degree, but an imperfect power of the third degree; that is, it is a perfect square but not a perfect cule. Indeed numbers are seldom perfect powers of more than one degree. 16 is a perfect power of the 2d and 4th degrees; 64 is a perfect power of the 2d, 3d and 6th degrees. 569. Every root, as well as power of 1, is 1. (Art. 562. Obs. 4.) Thus, (1)2, (1)3, (1)o, and √1, √1, √1, &c., are all equal. 3 6 PROPERTIES OF SQUARES AND CUBES. 570. The properties of numbers in general, have already been given. The following pertain to square and cubic numbers. 2 2 1. The product of any two or more square numbers, is a square; and the product of any two or more cubic numbers, is a cube. Thus 2X3-36, the square of 6; and 2a ×3°=216, is the cube of 6. 3 2. If a square number is divided by a square, the quotient will be a square. Thus, 144÷9-16, which is the square of 4. 3. If a square number is either multiplied or divided by a number that is not a square, neither the product nor quotient will be a square. 4. If you double the number of times a number is taken as a factor, it will not produce double the product, but the square of it. Thus, 3x3=9, and 3×3 X3X3=81, and not 18. 5. The product of two different prime numbers cannot be a square. 6. The product of no two different numbers, which are prime to each other, will make a square, unless each of those numbers is a square. 7. The square and cube of an even number are even; and the square and cube of an odd number are odd. (Art. 161. Prop. 6, 10.) Hence, QUEST.-567. What is a perfect power? What is a rational number? 588. An imperfect power? A surd? Obs. Are numbers ever perfect powers of one degree and imperfect powers of another degree? 5C9. What are all roots and powers of 11 |