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Single Position ? After having ascertained the result of the operation, how will you proceed? How will you first begin an operation in Double Position ? After having obtained the first error, how will you proceed? When you have obtained the second error, what is to be done? What will you do after you have multiplied the second supposition by the first error, and the first supposition by the second error ? When you have ascertained whether the errors are both of the same kind, how do you proceed? If they are not of the same kind, how will you proceed? What is the rule for Double Position ?
8. A laborer was hired for 60 days upon this condition : that for every day he wrought, he should receive 75 cents, and for every day he was idle, he should forfeit 37} cents; at the ex. piration of the time he received D18; how many days did he work, and how many was he idle ?
Ans. worked 36 days, was idle 24 days. 9 Two persons, A. and B., have the same income; A.'saves of his yearly; but B., by spending D150 per annum more than A., at the end of 8 years finds himself D400 in debt; what is their income, and what does each spend per annum ?
Ans. income D400; A spends D300; B. D450.
INVOLUTION, OR THE RAISING OF POWERS,
Teaches the method of finding the powers of numbers.
A power is the product arising from multiplying any given number into itself continually a certain number of times ; thus, 2x2=4, is the second power or square of 2; 2x2x2=8, the third power or cube of 2; 2x2x2x2=16, the fourth power of 2, &c. The number which denotes a power is called its index. If two or more powers are multiplied together, their product is that power whose index is the sum of the exponents of the factor ; thus 2x2=4, square of 2; 4X4=16, fourth
power of 2; and 16 x 16=256, eighth power of 2, &c. often denoted by a figure placed at the right and a little above che number, which figure is called the index or exponent of that power (thus, 22, 33), and is always one more than the number of multiplications to produce the power, or is equal to the num.
ber of times the given number is used as a factor in producing the power. In producing the square of 2, there is only one mul. tiplication, or two factors ; in producing the cube, there are two multiplications or three factors, 33 x 3 x 3=27, &c. This subject will be more fully illustrated in progression.
Multiply the given number or first power continually by itself till the number of multiplications be 1 less than the index of the power to be found, and the last product will be the power required. Fractions are multiplied by taking the products of their numerators, and of their denominators; they will be involved by raising each of their terms to the power required, and if a mixed number be proposed, either reduce it to an improper fraction, or reduce the vulgar fraction to a decimal, and proceed by the rule.
1. Required the third power or cube of 35.
Ans. 35 X 35 X 35=42875. 2. What is the 5th power of. 7 ?
Ans. 16807. 3. What is the 5th power of 9?
Ans. 59049=95. 4. What is the 5th power of ? 5. What is the 4th power of .045?
Here we see that in raising a fraction to a higher power,
crease its value.
6. What is the third power of .263 ? Ans. .018191447. 7. What is the eighth power of 1?
Ans. 36 8. What is the square of 60 ?
Ans. 3600. 9. What is the square of } ?
Ans. 10. What is the square of .01 ?
Ans. .0001. 11. What is the cube of 21 ?
Ans. 112 12. What is the ninth power of 747 ? 13. What is the seventh power of 298.75 ?
What is a power? How do you raise a number to any pow. er? What is the rule !
EVOLUTION, OR THE EXTRACTION OF ROOTS.
EVOLUTION, or the extraction of roots, properly belongs to mathematics, and without a knowledge of that science, it will require strict attention and close application to arrive at any degree of perfection in the use and principles of those rules. The most correct and convenient method of extracting the roots of the several powers, particularly those of the higher order, is by logarithmic tables, as far preferable to any rules that can be given in common arithmetic.
The root of a number, or power, is such a number as, being multiplied into itself a certain number of times, will produce that number or power, and is denominated the square, cube, biquadrate, &c., or 2d, 3d, and 4th root, accordingly as it is, when raised to the 2d, 3d, and 4th power, equal to that power. Thus 4 is the square root of 16, because 4X4=16 : and 4 is the cube root of 64, because 4 X 4X4=64 ; and 4 is the fourth root or biquadrate-of 256, because 4X4 X4,X4=256, &c. The roots are proportional, but their proportion is different from simple or compound proportion; the raising of powers increase in a uniform ratio, but this will not always--indeed but seldomoccur in the extraction of roots. Although there is no number of which we can not find any power exactly, yet there are many numbers of which precise or exact roots can never be determined; but by the use of decimals we can approximate toward the root to any assigned degree of accuracy; those roots are called surds ; and those which are perfectly accurate, rational roots ; surd roots sometimes have their origin in circulating decimals, or vulgar fractions. As few numbers are complete powers, surds must very often occur in arithmetical operations, but the result can be obtained nearly by continuing the extraction of the root.
1. SEPARATE the given number into periods of two figures caeh, beginning at the right hand or place of units.
2. Begin at the left hand, and find the quotient root in that
sriod, and place it on the right of the given sum in the quo1 ynt, and its square under said period, which subtract from the number above.
3. Then bring down the next period of 2 figures, and place it on the right of the remainder, as in division, and this forms a new dividend.
4. Now double this figure or root in the quotient, and place it on the left of the new dividend for a divisor.
5. Then consider how often the divisor is contained in the dividend, omitting the last figure, and place the result on the right of the root in the quotient, and then place this figure on the right of the number produced by doubling for a divisor, and multiply as in division, until the periods are all brought down.
For Decimals.—When decimals occur in the given nuniber, it must be pointed both ways from the decimal point, and the root must consist of as many figures, of whole numbers and decimals respectively, as there are periods of integers or decimals in the given number. When a decimal alone is given, annex one cipher, if necessary, so that the number of decimal places shall be equal; and the number of decimal places in the root will be equal to the number of periods in the given decimal
For Vulgar Fractions.-1. Reduce mixed numbers to im proper fractions, and compound fractions to simple ones, and then reduce the fraction to its lowest terms.
2. Extract the square root of the numerator and denominator separately, if they have exact roots; but if they have not, reduce the fraction to a decimal, and then extract the root, as above, &c. Proof: square
the root and add in the remainder.
Illustration.--A square number can not have more places of figures than double the places of the root, and at least but ono less. А
square is a figure of four equal sides, each pair meeting perpendicularly, or a figure whose length and breadth are equal. As the area, or number of equal feet, inches, &c., in a square, is equal to the products of the two sides, which are equal, the second power is called the square. Let the following figure represent a board one foot square, and one inch in. thickness, which being sawn or cut into square or solid inches, will make 144 inches, or 144 blocks one inch square ; and the square root of 144 is 12 ; becausé 12 x12=144, which in this case will be inches=one side of the square.
Thus, 154682(393 root. 3932
Explanation. 3X3 9
First seek the
square of the 3x2=69)646
first period (15), 621 3537
is 9, it can not 39 X2=783)2582
be 4, because 2349 154449
233 rem. is more than 15; 233 rem.
place the 3 in
154682 proof. the quotient and the 9 under the first period (15), which subtract and you have 6; then bring down the next period (46), which place to the right of the remainder (6); now double the quotient figure, 3 x2=6, and place it in the divisor, one place to the left ; now consider how many times 6 in 64, which for trial call 9; write 9 in the quotient, and to the right of 6 in the divisor, which makes the divisor 69; then multiply 69 by 9, and place the result under the dividend ; subtract as before, and bring down the next period (82); now double the two quotient figures, 39+2=78, which place in the divisor; consider how many times 78 is contained in 258, suppose 3 times; place 3 in the quotient and in the divisor, as before ; then multiply the divisor by the last quotient figure, and write the result as above directed, and the remainder is 233; which by annexing ciphers may be continued. 2. What is the square root of 74770609 ?
Ans. 864% 3. What is the square root of 54990.25 ?
234.5. 4. What is the square root of 3271.4007 ?
57.19 te 5. What is the square root of 14876.2357 ?. 121.968175. 6. What is the square root of 96385103 ?