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. MENTAL EXERCISES. Ex. 1. Resolve 25 into two equal factors. Solution. 25=5x5. 2. Resolve 8 into three equal factors. Solution. 8=2x2x2. 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 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 evolution? 9 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 X2 X2=8. The cube root 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 ? Solution. The square root of the numerator 9, is 3; and the square root of the denominator 16, is 4. Therefore & is the square root of j; for åxå=16 12. What is the square root of į? Ans. $. 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 V4, or 4: denotes the square or 2d root of 4 ; 27, or 27+ denotes the cube or 3d root of 27; V16, or 167 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 v is placed before the given number. Thus the square root of 25 is written V 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? EXERCISES FOR THE SLATE. 17. Express the cube root of 45 both ways. 18. Express the cube root of 64 both ways. Of 125. 19. Express the fourth root of 181 both ways. Of 576. 20. Express the 5th root of 32 ; the 6th root of 64. 21. Express the 7th root of 84; the 8th root of 91 ; the 9th root of 105; the 10th root of 256. 22. Express the cube root of 576 ; the fourth root of 675; the fifth root of 1000; the twelfth root of 840. 347, 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, (the cube root of 27,) are rational numbers. 348. A number which cannot be resolved into equal factors, or whose root cannot be exactly extracted, is called an imperfect 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+; 6.7+, &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 cube. 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 and 5th degrees. 349. Every root, as well as every power of 1, is 1. (Art. 342.) Thus (1)2, (1), (1)®, and vi, oi, oi, &c. are all equal. Quest.-347. What is a perfect power? What is a rational number? 348. What is an imperfect power? What is a surd? Obs. Are num. bers ever perfect powers of one degree and imperfect powers of another degree? Are they often perfect powers of more than one degree ? 349. What are all roots and powers of 1? 4 yards. EXTRACTION OF THE SQUARE ROOT. 350. To extract the square root, is to resolve a given slumber into two equal factors; or, to find a number which being multiplied into itself, will produce the given number. (Art. 344. Obs.) Ex. 1. What is the side of a square room which contains 16 square yards ? 4 yards. Solution. Let the room be represented by the adjoining figure. It is divided into 16 equal squares, which we will call square yards. Since the room is square, the question is simply this : What is the square root of 16 ? Now if we resolve 16 into two equal factors, each of those factors will be 4X4=16 yards. the square root of 16. But 16=4 X4. The square root of 16, therefore, is 4. 2. What is the length of one side of a square room which contains 576 square feet? Operation. Since we may not see what the 576(24 root of 576 is at once, as in the last 4 example, we will separate it into periods 44)176 of two figures each, by putting a point 176 over the 5, and also over the 6; that is, over the units' figure and over the hundreds. This shows us that the root is to have two figures; (Art. 342. Obs 2 ;) and thus enables us to find the root of part of the number at a time. Now the greatest square of 5, the left hand period, is 4, the root of which We place the 2 on the right hand of the number for the first part of the root; then subtract its square from 5, the period under consideration, and to the right of the remainder bring down 76, the next period, for a dividend. To find the next figure in the root, we double the 2, the part of the root already found, and placing it on the left Quest.-350. What is it to extract the square root of a number? is 2. ure. of the dividend for a partial divisor, we find how many times it is contained in the dividend, omitting the right hand fig Now 4 is contained in 17, 4 times. Placing the 4 on the right of the root, also on the right of the partial divisor, we multiply 44, the divisor thus completed, by 4, the last figure in the root, and subtracting the product 176 from the dividend, find there is no remainder. The answer therefore is 24. Note.-Since the root is to contain two figures, the 2 stands in teps' place; hence the first part of the root found is properly 20; which being doubled, gives 40 for the divisor. For convenience we omit the cipher on the right; and to compensate for this, we omit the right hand figure of the dividend. This is the same as dividing both the divisor and dividend by 10, and therefore does not alter the quotient. (Art. 88.) Proof. 24=2 tens, or 20+4 units. 24-2 20+4 80+16 66 400+80 ILLUSTRATION BY GEOMETRICAL FIGURE. 4ft. 20ft. 20ft. Let the large square A 20ft. H B ABCD, represent the room in the last example; then the square DEFG will be the greatest E F square of the left hand. period, the root of which is 20 ft., and 20 x 20= 400, the number of feet in its area. (Art. 163.) But this square 400 ft. taken from 576 ft. leaves a remainder of 176 ft. D 20ft. G C Now it is plain if this remaining space is all added to one side of this square, its sides would become unequal; consequently it would cease to be a square. (Art. 153. Obs. 1.) But if it is equally enlarged on two sides it will obviously continue to be a Quest.-Note. What place does the first figure of the root occupy in the example above? Why is the right hand figure of the dividend omitted ? |