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CASE I.

$ 220. To extract the cube root of a whole num. ber.

RULE.

I. Point off the given number into periods of three figures each, by placing a dot over the place of units, a second over the place of thousands, and so on to the left: the left hand period will often contain less than three places of figures.

II. Seek the greatest cube in the first period, and set its root on the right, after the manner of a quotient in division. Subtract the cube of this figure of the root from the first period, and to the remainder bring down the first figure of the next period, and call the number the dividend.

III. Take three times the square of the root just found for a divisor and see how often it is contained in the dividend and place the quotient for a second figure of the root. Then cube the figures of the root thus found, and if their cube be greater than the two first periods of the given number, diminish the last figure, but if it be less subtract it from the two first periods, and to the remainder bring down the first figure of the next period, for a new dividend.

IV. Take three times the square of the whole root for a new divisor, and seek how often it is contained in the new dividend : the quotient will be the third figure of the root. Cube the whole root and subtract the result from the first three periods of the given number, and proceed in a similar way for all the periods.

Ez. 1. What is the cube root of 99252847 ?

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4* *3=48)352 dividend Two first periods

99252 (46)=46 X 46 X 46= 97336

3x (46) =6348 ) 19168 2d dividend The first three periods 99252847. (463)

=99252847.

Ans. 463. 2. What is the cube root of 389017?

Ans. 73. 3. What is the cube root of 5735339 ?

Ans. 179. 4. What is the cube root of 32461759 ?

Ans. 319. 5. What is the cube root of 84604519 ?

Ans. 439. 6. What is the cube root of 259694072 ?

Ans. 638. 7. What is the cube root of 48228544 ?

Ans. 364. 8. What is the cube root of 27054036008 ?

Ans. 3002.

CASE II.

0 221. To extract the cube root of a decimal frac, tion.

RULE.

Annex ciphers to the decimal, if necessary, so that it shall consist of 3, 6, 9, 8-c. places. Then place the first point over the place of tenths, the second oder the place of ten thousandths, and so on over everu

third place to the right ; after which extract the root as in whole numbers.

Note 1. There will be as many decimal places in the root as there are periods in the given number.

Note 2. The same rule applies when the given number is composed of a whole number and a decimal.

Note 3. If in extracting the root of a number there is a remainder, after all the periods have been brought down, periods of ciphers may be annexed. by considering them as decimals.

EXAMPLES.
1. What is the cube root of ,157464 ?

Ans. ,54.
2. What is the cube root of ,870983875 ?

Ans. ,955. 3. What is the cube root of 12,977875 ?

Ans. 2,35. CASE III. $ 222. To extract the cube root of a vulgar fraction.

RULE. I. Reduce compound fractions to simple ones, mixed numbers to improper fractions, and then reduce the fraction to its lowest terms.

II. Then extract the cube root of the numerator and denominator separately, if they have exact roots ; but if either of them has not an exact root, reduce the fraction to a decimal, and extract the root as in the last Case.

EXAMPLES. 1. What is the cube root of ff. ?

Ans. If

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QUESTIONS. $213. What is Evolution ? What does it teach?

§ 214. What is the square root of a number? What is the cube root of a number 1 Make the sign denoting the square root? How do you denote the cube root ?

$ 215. What is required when we wish to extract the square root of a number? What number is greater than the square of either of the significant figures ?

$ 216. How do you extract the square root of a whole number ? How many figures will there be in the root ? If the given number has not an exact root, what may be done ?

$ 217. How do you extract the square root of a decimal fraction? When there is a decimal and whole number joined together ?

§ 218. How do you extract the square root of a vulgar fraction ?

$219. What is required when we are to extract the cube root of a number? $ 220. How do you extract the cube root of a whole number?

221. How do you extract the cube root of a decimal frac. tion? How many decimal places will there be in the root ? Will the same rule apply when there is a whole number and a decimal ? In extracting the root is there be a remainder, what may be done?

$ 222. How do you extract the cube root of a vulgar frac. tion ?

ARITHMETICAL PROGRESSION. $ 223. If we take any number, as 2, we can, by the continued addition of any other number, as 3 form a series of numbers; thus,

2, 5, 8, 11, 14, 17, 20, 23, &c. in which each number is formed by the addition of 3 to the preceding number.

This series of numbers may also be formed by subtracting 3 continually from the largest number; thus,

23, 20, 17, 14, 11, 8, 5, 2. A series of numbers formed in either way is call. ed an Arithmetical Series, or an Arithmetical Progression; and the number which is added or subtracted is called the common difference.

When the proportion is formed by the continued addition of the common difference, it is called an ascending series; and when it is formed by the subtraction of the common difference, it is called a descending series ; thus,

2, 5, 8, 11, 14, 17, 20, 23, is an ascending series. 23, 20, 17, 14, 11, 8, 5, 2. is a descending series.

The several numbers are called the terms of the progression : the first and last terms are called the extremes, and the intermediate terms are called the

means.

§ 224. In every arithmetical progression there are five things which are considered, any three of which being given or known, the remaining two can be determined. They are

1st the first term,
2d the last term,
3d the common difference,
4th the number of terms,

h the sum of all the terms.

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