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Continue the operation until the root is completed, or the approximation carried as far as is desired.
In order to avoid error, observe carefully the value of each root figure and each product. Thus, if the first root figure is hundreds, the number in the second column will be hundreds,-in the third, ten thousands,-in the fourth, millions, &c.
EXAMPLES FOR THE BOARD.
What is the third root of 205692449327 ?
104483109 c. d.
The complete divisors are marked c. d., the trial divisors, t. d. The figures at which the new additions commence, are marked (1), (2). The partial dividends by which each root figure is determined, are distinguished by a comma. They always terminate with the first figure of the period that is annexed. The abbreviations, thous., mill., &c., show the value of the figures against which they are placed.
[The extraction of the square root, and the solution of equations of the second power (of which examples are given on p. 125), afford very ready and convenient applications of this rule. The determination of the first root figure in the higher powers, would frequently be difficult, without the aid of Table V. In the solution of many Algebraical Equations, even this table affords no assist. ance, but we are obliged to rely upon trial, for the first figure of the root, which being ascertained, the succeeding figures will be easily found.]
(2) 3052 tenths 3727104
The additions to the left hand column may be made mentally, and thus shorten the labor. There are other abbreviations, for which the student is referred to the Chapter on Numerical Approximations.
The first root figure in each of the following examples, may be found by the table of Powers and Roots.
1. Extract the square root of 350026681.
2. Extract the square root of 3; 5; 6.5.
3. Extract the cube root of 2924207.
4. Extract the cube root of 13; 12.5.
5. Extract the fifth root of 65.7748550151.
6. Extract the 7th root of 1.246688292353624506368.
To show the universal application of the rule, we will solve a question containing many of the powers of a number. At the head of the columns we write the coefficients* of all the powers, from the highest to the lowest, substituting a 0 when any power is wanting. The first coefficient is then multiplied by the root figure, and the product ad
* A coefficient is a figure indicating the number of times any term is employed. Thus, in 7 times the 5th power, 7 is the coefficient of the 5th power.
ded to the second coefficient, the product of this sum by the root figure, added to the third coefficient, and so on.
EXAMPLE FOR THE BOARD.
7 times the 5th power, minus 2 times the 3d power, plus 5 times the second power of a certain number, is equal to 1405569. 53125. What is the number?*
We commence with writing in order the coefficients, 7 for the 5th power, 0 for the 4th power,-2 for the 3d power, 5 for the 2d power, and 0 for the first power. Then, as the number has two integral periods, the first figure of the root will be tens. Finding by trial that 1 is the first root figure, we add 10×7 to 0; 10 X 70 to -2; &c., as in the following solution:
The equation stated algebraically, would be, 7x5 — 2x3 + 5x2 = 1405569.53125.
The first root figure in the following examples must be found by trial.
7. Five times the third power, plus three times the second power of a number, added to twice the number, make 9349968. What is the number?
8. A man being asked his age, replied: "If 3 times the square of my age be added to 5 times my age, the sum will be 2668." What was his age?
9. 7 times the 4th power +2 times the third power-13 times the second power of a certain number=2120912536. What is the number?
10. The 5th power of a certain number, diminished by 23 times the number, equals 14025514846. What is the number?
11. What is that number whose 5th power +5 times its 4th power-631 times its second power=42629137739 ?
12. What is that number 5 times whose 4th power+25 times its 3d power+125 times its 2d power=1364615. 9375?
13. What is the number of miles from New York to Baltimore, if 13 times its 4th power+79 times its 3d power-93 times its 2d power,=17848292769 ?
14. The 6th power of a certain number exceeds its 2d power by 65944160560800. What is the number?
15. Extract the cube root of 3; the 5th root of 4; the 7th root of 5.
16. What is the value of x, if 5x5 +3x2+17x=16035686129037 ?
When two equal quantities are connected by the sign of equality, as in the above example, the whole is called an EQUATION.
A QUADRATIC EQUATION, is one in which the highest power of the unknown quantity is the square.
17. Find the value of x in the quadratic equation 3x2 +7x=780.
18. Find the root of the quadratic equation 2x2+5x= 37125.
19. What is the value of x, if 9x2+13x=58006 ?
20. What is the value of x in the quadratic equation, 4x2+x=159.3275?
ARITHMETICAL PROGRESSION-OR, EQUIDIF-
A series of numbers, in which the successive terms increase or diminish uniformly by the same number, is called an ARITHMETICAL PROGRESSION,-PROGRESSION BY DIFFERENCE, or EQUIDIFFERENT SERIES. The difference between the successive terms, is called the common difference. The first and last terms of the series are called the extremes; the other terms the means.
Thus, in the ascending series,
2, 4, 6, 8, 10, 12, 14, 16,
the extremes are 2 and 16, and the common difference is 2. In the descending series,
20, 16, 12, 8, 4, 0,
the extremes are 20 and 0, and the common difference is 4. Any three of the five following things being given, the other two may be found:
1. The first term.
2. The last term.
3. The number of terms.
4. The common difference
5. The sum of all the terms.
One of the extremes, the common difference, and the number of terms being given, to find the other extreme and the sum of all the terms.
What is the tenth term of an ascending series, the first term being 1 and the common difference 4?
The second term will evidently be 1+4; the 3d, 1+2×4; the 4th, 1+3×4, and so on to the 10th, which is 1+9x4 or 37.
What is the sum of the first ten terms of the above series?
To obtain a rule for finding the sum, we will invert the whole series, and write it under itself. In this manner we shall evidently obtain twice the sum of the series. And we may moreover observe, that the sum of the extremes is equal to the sum of