342. a. The square of any number which consists of two figures is equal to the square of the tens, added to twice the product of the tens into the units, added to the square of the units. Note.-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. 7.) 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. 7;) 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 1X1=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 10X10=100, has one figure less than both factors. Hence, 2. Any square number 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. 12. What is the square of 45? 13. What is the cube of 5? 50? 75? 100 ? 540? Of 8? 10? 12? 60? 14. What is the fourth power of 3? Of 4? 16? 20? 15. What is the fifth power of 2? Of 3 4 5 6? Of } ? ? ? ? ? Of 10 Q? Of 3? 5? 103 ? 16. What is the square of? 17. What is the cube of? Of ? Of ? 18. What is the square of 2? 19. What is the square of 1.5? Of 3.25? Of 10.25? EVOLUTION. 343. 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, QUEST.-342. a. What is the square of any number consisting of two figures equal to ? Note. How many figures are there in the product of any two factors? How many figures will the square of a number contain? The cube? 343. When a number is resolved into any number of equal factors, what is each of those factors called? 344. A root of a number is a factor, which, being multiplied into itself a certain number of times, will produce that number. OBS. 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; when resolved into four equal factors, each factor is called the fourth root; &c. Hence, the name of the root expresses the number of equal factors into which the given number is to be resolved. Thus the second or square root, shows that the number is to be resolved into two equal factors; the third or cube root, into three equal factors; the fourth root, into four equal factors, &c. Thus, The square root of 16 is 4; for 4X4=16. The cube root of 27 is 3; for 3×3×3=27. The fourth root of 16 is 2; for 2×2×2×2=16, &c. MENTAL EXERCISES. Ex. 1. Resolve 25 into two equal factors. 2. Resolve 8 into three equal factors. 345. 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. 341.) 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 correlative terms. If one number is a power of another, the latter is a root of the former. Thus 27 is the cube of 3; and 3 is the cube root of 27. 2. The learner will be careful to remember, that Ans. 4. 3. What is the square root of 16? QUEST.-344. What then is a root? Obs. What does the name of the root express? What does the square root show? The cube root? The fourth root? 345. What is evolution? Obs. Of what is it the opposite? Into what are numbers resolved in subtraction? In division? In evolu 5. What is the square root of 64? Of 81? Of 100? Of 121? Of 144? 6. What is the cube or third root of 8? Solution. If we resolve 8 into three equal factors, each of these factors is 2: for 2×2×2=8. of 8 therefore is 2. 7. What is the cube root of 27 ? 8. What is the cube root of 64? 9. What is the cube root of 125? 10. What is the fourth root of 16? 11. What is the square root of 16? The cube root Solution. The square root of the numerator 9, is 3; and the square root of the denominator 16, is 4. 25 fore is the square root of; for 36 x=1 Ans. §. Of 25 ? Of 64 ? Ans.. Of 27? 4 There 346. Roots are expressed in two ways; one by the radical sign (v) placed before a number; the other by a fractional index placed above the number on the right hand. Thus 1/4, or 4 denotes the square or 2d root of 4; † 27, or 27 denotes the cube or 3d root of 27; 16, or 16 denotes the 4th root of 16. 3 OBS. 1. The figure placed over the radical sign, denotes the root 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. QUEST.-346. In 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? 344. A root of a number is a factor, which, being multiplied into itself a certain number of times, will produce that number. OBS. 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; when resolved into four equal factors, each factor is called the fourth root; &c. Hence, the name of the root expresses the number of equal factors into which the given number is to be resolved. Thus the second or square root, shows that the number is to be resolved into two equal factors; the third or cube root, into three equal factors; the fourth root, into four equal factors, &c. Thus, The square root of 16 is 4; for 4X4=16. The cube root of 27 is 3; for 3×3×3=27. The fourth root of 16 is 2; for 2×2×2×2=16, &c. MENTAL EXERCISES. Ex. 1. Resolve 25 into two equal factors. 2. Resolve 8 into three equal factors. 345. 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. 341.) 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 correlative terms. If one number is a power of another, the latter is a root of the former. Thus 27 is the cube of 3; and 3 is the cube root of 27. 2. The learner will be careful to remember, that 3. What is the square root of 16? Ans. 4. QUEST.-344. What then is a root? Obs. What does the name of the root express? What does the square root show? The cube root? The fourth root? 345. What is evolution? Obs. Of what is it the opposite? Into what are numbers resolved in subtraction? In division? In evolu 5. What is the square root of 64? Of 81? Of 100? Of 121? Of 144? 6. What is the cube or third root of 8? Solution. If we resolve 8 into three equal factors, each of these factors is 2: for 2×2×2=8. The cube root There Solution. The square root of the numerator 9, is 3; and the square root of the denominator 16, is 4. fore is the square root of; for 4×116. 9 Ans.. ? Of 25? 49 Of 64 ? 100 12. What is the square root of ? 27. Ans. 1. 64 346. Roots are expressed in two ways; one by the radical sign (v) placed before a number; the other by a fractional index placed above the number on the right hand. Thus 14, or 4 denotes the square or 2d root of 4; √ 27, or 27* denotes the cube or 3d root of 27; 16, or 16 denotes the 4th root of 16. 3 OBS. 1. The figure placed over the radical sign, denotes the root 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. QUEST.-346. In 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? |