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Or, Take from the preceding Table, the amount of one pound, or one dollar, as the case may be, for the given number of years, and at the given rate per cent., and multiply it by the given principal, and it will give the amount as before.
years, at 6
1. What will £400 amount to in 4 per annum, compound interest? 400x1,06x1,06x1,06x1,06=£504,99+ or £504 19s.
9d. 2,4qrs. + Ans. Or, by the Table. Tabular amount of £1=1,26247 Multiply by the principal
Whole amount £504,98800 2. What is the compound interest of $555 for 14 years, at 5 per cent. ?
Ans. 8543,86cts. + Note.—Any sum of money,
per annum, simple interest, will double in 163 years; but at 6 per cent. per annum compound interest, it will double in 11 years and 325 days, or 11,889 years.
ANNUITIES AT COMPOUND INTEREST.
CASE 1.- To find the amount of an annuity, fc. Rule.-Raise the amount of $1, or £1, at the given rate per cent., for one year, to that power denoted by the given number of years; subtract unity or 1 from this product; multiply the remainder by the given annuity; divide this last product by the ratio made less by unity or 1 ; and the quotient will be the amount sought.
1. If $250, yearly pension, be foreborne 7 years, what will it amount to, at 6 per cent. per annum compound interest ? 1,06 1,06 x 1,06 x 1,06x1,06 1,06 x 1,06—1,x250
$2098,45c. 9m. + Ans.
2. If a salary, or an annuity, of £100 per annum, runs on unpaid for 6 years, at 5 per cent. compound interest,
6 what is the amount due at the end of that period ?
Ans. £680 3s. 9 d. ,63. CASE 2.-To find the present worth of an Annuity, foc.
Rule. Raise the amount of 81, or £1, at the given rate per cent., for 1 year, to that power denoted by the given number of years; divide the given annuity by this product; subtract its quotient from the given annuity; divide the remainder by the ratio made less by unity or 1 ; and the quotient will be the present worth sought.
EXAMPLES. 1. What is the present worth of a salary of $300, to continue 5 years, at 5 per cent. compound interest ? 300
=235,0578499405. + 1,05 x 1,05 1,05 x 1,05 x 1,05
300—235,0578499405 8 Then,
=1298,84 3+Ans. 1,05–1, 2. What is the present worth of £30 per annum, to continue 7 years, at 6 per cent. compound interest ?
Ans. £167 9s. 5d. +
INVOLUTION. INVOLUTION is the continual multiplication of a number into itself; and the products thence arising, with the original number itself, are called the powers of that num-' ber.
Any number may itself be called a first power. If the first power be multiplied by itself, the product is called the second power, or square; if the square be multiplied by the first power, the product is called the third power, or cube ; if the cube be multiplied by the first power, the product is called the fourth power, or biquadrate, &c.
Thus 3 is the first power of 3. 3x3=9 is the second power of 3. 3x3x3=27 is the third power of 3.
3x3x3x3=81 is the fourth power of 3 &c. &c. And in this manner is formed the following table of powers.
Table of the SQUARES and CUBES of the nine digits. Roots. 1 2 3 4 5 61 71 81 9
Cubes. 11 8|27|64| 125 2161 343 5121 729
EXAMPLES. 1. What is the 6th power of 8?
Note.—The nom8 the root, or 1st power.
height of the power
is called the index, 64=2d power, or square.
or exponent of that 8
power; so the 2d
power of 3 may be 512=3d power, or cube. denoted by 32, the 3d 8
by 33, the 4th by 3*,
&c.; the 6th power 4096=4th power,or biquadrate. of 8 by 86, &c. .
32768=5th power, or sursolid.
262144–6th power, or square cube.
Ang. 86. 2. What is the 7th power of ?
Aps. 128 3. What is the 5th power of or 11? Ans. 59047 4. What is the fourth power of ,277 Ans. ,00531441.
EVOLUTION, OR EXTRACTION OF ROOTS.
When the root of any power is required, the business of finding it is called the extraction of the Root.
The root is that number, which by a continual multiplication into itself, produces the power which is given to be extracted.
Though every number will produce a perfect power by involution, yet there are many numbers, the precise roots of which can never be determined. By the help of decimals, however, we can approximate towards the root, to any assigned degree of exactness.
The roots which approximate, are called surd roots, and those which are perfectly accurate, are called rational roots.
TO EXTRACT THE SQUARE ROOT. Any number multiplied into itself, produces a square. The extracting of the square root, is only finding a number, which, being multiplied into itself, shall produce the given number.
RULE.-1. Distinguish the given number into periods of two figures each, by putting a point over the place of units, another over the place of hundreds, and so on over every second figure; and if there be decimals, point them in the same manner, from units towards the right hand; which points show the number of figures the root will consist of.
2. Find by the table or trial the greatest square number in the first, or left hand period, place the root of it at the right hand of the given number, (after the manner of a quotient in division,) for the first figure of the root; and set the square number under the period, subtract it therefrom, and to the remainder bring down the next period for a dividend.
3. Place the double of the root already found on the left hand of the dividend for a divisor.
4. Consider what figure must be annexed to the divisor, so that if the result be multiplied by it the product may be equal to, or the next less than, the dividend, and it will be the second figure of the root.
5. Subtract that product from the dividend, and to the remainder bring down the next period for a new dividend.
6. Find a divisor as before, by doubling the figures already in the root; and from these find the next figure of the root as in the last article ; and so on through all the periods to the last.
Or, to facilitate the foregoing operation, when a period is brought down to a remainder, and a dividend thus formed, in order to find a new figure in the root, divide said dividend, (omitting the right hand figure thereof,) by double the root already found, and the quotient will commonly be the figure of the root sought, or, being made less by one, or two, will generally give the next figure sought.
To extract the square root of a Vulgar Fraction. First prepare all vulgar fractions by reducing them to their lowest terms, both for this and all other roots. Then,
1. Take the root of the numerator and that of the denominator for the respective terms of the root required ; and this is the best way if the denominator be a complete power. But if not,
2. Multiply the numerator and denominator together; take the root of the product; this root, being made the numerator to the denominator of the given fraction, or the denominator to the numerator of it, will form the fractional root required.
3. Or reduce the vulgar fraction to a decimal, and extract its root.
EXAMPLES. 1. Required the square root of 6749604.
6749604(2598 Ans. The root exactly without a 4
remainder ; but when the periods be
longing to any given number are all 45)274 exhausted, and still leave a remain5)225 der, the operation may be continued
at pleasure, by annexing periods of 509)4996 ciphers, &c.-Roots are often denot9)4581 ed by writing ✓ before the power,
with the index of the root within or 5188)41504 over it, save the index of the square 8)41504 root, which is ever understood: so ✓
64 is the 2d or square root of 64 ; 3 2. Required the 64 the 3d or cube root of 64; * v 64 square root of 739,4 the 4th root of 64.
When the square root of a number
is wanted to many places, the work. 47)339 may be much abridged. Find half 7)329 the root by the rule; then to get the
rest, annex to the last remainder as 541) 1040
many ciphers as you need, and divide • 1) 541 it by the double of the root before