11. What is the fifth root of 243? Of 1024? Of 3125? When it is required to find the fourth root of a large number, find first, the second or square root, and then find the square root of this square root, and it will be the fourth root required. 12. What is the fourth root of 6561? Of 65536? Of 390625? We may find the sixth root by taking the third root of the square root, &c. Any root may be found directly, by the following rule: RULE FOR EXTRACTING ANY ROOT.-1st. Divide the number into periods consisting of as many figures as there are units in the corresponding power, commencing at the right hand. The first period on the left, may consist of any number of figures less than the number of units in the corresponding power. 2nd. Find by trial the whole number which is nearest the root of the first period on the left, and which is less than the exact root of this period. 3rd. Subtract the power of this whole number from the first period, and bring down the next period to the right hand of the remainder. 4th. Raise the root found, to the power next below the one corresponding to the root, and multiply it by the exponent of the corresponding power, for a divisor. 5th. Divide the remainder with the first figure of the period brought down annexed, and place the quotient at the right of the figure found. 6th. Raise the whole root thus found, to the corresponding power, and subtract the power from the first two periods. Bring down the next period of figures, and proceed as before, until all the figures in the root are found. 13. What is the fourth root of 50625? 14. What is the fifth root of 9765625? Of 390625? Of 5153852? SECTION XLVI. On Cubic Equations. 1. Find the values of x in the following equation: (x — 5) (x2 — 7 x + 12) = 0 (1) In this equation we have two factors in the first member; either may be equal to 0. Put the first = 0, and we have x-5=0 x=5 Put the second equal to 0, and we have x2-7x+12=0 x2-7x=- -12 x2-7x+12= X- · 31 = + 1, or This equation gives three values of x, 3, 4 and 5. Putting the second factor = 0, gave an affected equation, which may be changed to another form, as before explained, (x-3)(x-4)= x2-7x+12 Substituting this value of x2-7x+12 in equation (1,) we have (x — 5) (x —3) (x —4) = 0 We have here three factors, the continued product of which is equal to 0. Either may be = 0. Putting the first Putting the second = 0, gives x=3.. Putting the third Equation (1) may be changed to another form, by multiplying as the signs indicate. This is called a cubic equation, or an equation of the third degree; because, it contains the third power of x, the unknown quantity. It is the same as equation (1,) and gives the same values of x. 2. Find the values of x in the following equations, and change the forms of the equations: Equation (2) may be changed to a form which shall contain only the first and third powers of the unknown quantity, by substituting for x, y and of the coefficient of the second power of x with the contrary sign. In equation (2) x3- 12x2 + 47x-124=0. Assume y+4= x; and substituting, we have (y + 4)3 → 12 (y + 4)2+ 47 (y +4)=60 y3 + 12 y2+48y+64-12 y2-96 y-19247y+188 €60 y3 — y = 64 In this way, every cubic equation may be reduced to one which will not contain the second power of the unknown number. Change the following equations as in the preceding example. (1) 23-9x216 x = 120 120 x2 +63 x = 56 x2+18 x2+66 x = : 500 x2+5 x2-36 x = 20 SECTION XLVII. On Cubic Equations. A cubic equation, as it is obtained from a question, contains generally the third, second and first powers of the unknown number. The second power may be made to disappear, as is evident from the examples in the latter part of the last section. Hence every cubic equation may be reduced to one of the following forms: When the equation is reduced to one of these forms, we propose to show how the value of the unknown number may be obtained. 3. Given, (1) 23 -3x=2 to find the value of x. 1 Assume x = of the coefficient of x divided by u, plus u, 3 1 Substitute this value of x in place of x in equation (1,) and we have Or, ÷ + 3 (+u) + w−3 ( + u) =2 1 +u3 = 2 This reduced as an affected equation gives u = 1, x=+u=2. 4. Find the values of x in the following: น ·(1) x3—6x=9, (2) x3 — 12 x = 880, (3) x3 — 3 x — — 2. 5. What number is that, from the second power of which, if 18 be subtracted and the remainder multiplied by the number, the product will be 108? If the term which contains the first power of x is plus or positive, as in the forms (3) and (4) given above, we assume x=of the coefficient of x divided by u, —u. 6. Given, x+6x=88 to find the value of x. 7. Find the value of x in the following equations: 8. The third power of a number added to six times the number, is equal to 155. What is the number? 9. The third power of the number of dollars A had is equal to the number B had. Both together had 68 dollars. How many had each? 10. A had a square piece of land, out of which he reserved 4 square rods, and sold the remainder for as many dollars per square rod as there were rods in one side of the square. What was the length of one side of the square, supposing he received 48 dollars for it? By reducing the general equations (1,) (2,) (3,) (4,) we shall find a general value of x for each equation. 28; what is the value of x? +u, and substituting in equation (1,) we 11. If (1) 3. 3px: Assume x = have 22 + 3p (+u) + w—3p (2+u)= =28 |