3. What is the 8th root of 1099511627776?

Ans. 32.

4. What is the 6th root of 25632972850442049? Ans. 543. 5. What is the 9th root of 1.577635? NOTE.Extract the cube root of the cube root by the contracted method,

Ans. 1.051963+.

carrying the root in each operation to 6 decimal places only.

6. What is the 12th root of 16.3939?

Ans. 1.2624+.

Ans. 1.2950+.

7. What is the 18th root of 104.9617?

CASE II.

665. When the index of the required root is prime, or contains any other factor than 2 or 3.

To extract any root of a number is to separate the number into as many equal factors as there are units in the index of the required root; and it will be found that if by any means we can separate a number into factors nearly equal to each other, the average of these factors, or their sum divided the number of factors, will be nearly equal to the root indicated by the number of factors.

to the 6th power, and divide the given number, 308, by the result, and obtain 2.0272+ for a quotient; we thus separate 308 into 7 factors, 6 of which are equal to 2.31, and the other is 2.0272. As these 7 factors are nearly equal to each other, the average of them all must be a near approximation to the 7th root. Multiplying the 2.31 by 6, adding the 2.0272 to the product, and dividing this result by 7, we

find the average to be 2.2696, which is the first approximation to the required root. We next divide 308 by the 6th power of 2.2696, and obtain 2.253452+ for a quotient; and we thus separate the given number into 7 factors, 6 of which are each equal to 2.2696, and the other is 2.253452. Finding the average of these factors, as in the former steps, we have 2.267293, which is the 7th root of the given number, correct to 5 decimal places. Hence the following

RULE. I. Find by trial some number nearly equal to the required root, and call this the assumed root.

II. Divide the given number by that power of the assumed root denoted by the index of the required root less 1; to this quotient add as many times the assumed root as there are units in the index of the required root less 1, and divide the amount by the index of the required root. The result will be the first approxi

mate root required.

III. Take the last approximation for the assumed root, with which proceed as with the former, and thus continue till the required root is obtained to a sufficient degree of exactness.

NOTES.-1. The involution and division in all cases will be much abridged by decimal contraction.

2. If the index of the required root contains the factors, 2 or 3, we may first extract the square or cube root as many times, successively, as these factors are found in the index, after which we must extract that root of the result which is denoted by the remaining factor of the index. Thus, if the 15th root were required, we should first find the cube root, then the 5th root of this result.

EXAMPLES FOR PRACTICE.

1. What is the 20th root of 617?

SERIES.*

666. A Series is a succession of num .ers so related to each other, that each number in the succession may be formed in the same manner, from one or more preceding numbers. Thus, any number in the succession, 2, 5, 8, 11, 14, is formed by adding 3 to the preceding number. Hence, 2, 5, 8, 11, 14, is a series.

667. The Law of a Series is the constant relation existing between two or more terms of the series. Thus, in the series, 3, 7, 11, 15, we observe that each term after the first is greater than the preceding term by 4; this constant relation between the terms is the law of this series.

The law of a series, and the term or terms on which it depends being given, any number of terms of the series can be formed. Thus, let 64 be a term of a series whose law is, that each term is four times the preceding term. The term following 64 is 64 x 4, the next term 64 × 42, etc.; the term preceding 64 is 64 ÷ 4. Hence the series, as far as formed, is 16, 64, 256, 1024.

668. A series is either Ascending, or Descending, according as each term is greater or less than the preceding term. Thus, 2, 6, 10, 14, is an ascending series; 32, 16, 8, 4, is a descending series.

669. An Extreme is either the first or last term of a series. Thus, in the series, 4, 7, 10, 13, the first extreme is 4, the last, 13. 670. A Mean is any term between the two extremes. Thus, in the series, 5, 10, 20, 40, 80, the means are 10, 20, and 40.

*The treatise of the "METRIC SYSTEM," as presented at some length at the close of the editions of this book published previous to the year 1875, was of little value, since the system is scarcely any used in this country, and the symbols introduced were different from those authorized or in use.

Being frequently requested by teachers to add or substitute an article on Mensuration, as of much more practical value, the editor has carefully prepared and substituted such a treatise at the end of this book, and to avoid repetition and put in more condensed form, has embodied in it the "applications of the square and cube roots" that intervened between "Evolution" and "Series" of former editions. So much of the Metric System as is needful has also been added.

The article on “Mensuration" is essentially the same as that presented in the "Complete Arithmetic" of the "Shorter Course," and it is hoped will be entirely satisfactory.

671. An Arithmetical or Equidifferent Progression is a series whose law of formation is a common difference. Thus, in the arithmetical progression, 3, 7, 11, 15, 19, each term is formed from the preceding by adding the common difference, 4.

672. An arithmetical progression is an ascending or descending series, according as each term is formed from the preceding term by adding or subtracting the common difference. Thus, the ascending series, 7, 10, 13, 16, etc., is an arithmetical progression in which the common difference, 3, is constantly added to form each succeeding term; and the descending series, 20, 17, 14, 11, 8, 5, 2, is an arithmetical progression in which the common difference is constantly subtracted, to form each succeeding term.

673. A Geometrical Progression is a series whose law of formation is a common multiplier. Thus, in the geometrical progression, 3, 6, 12, 24, 48, each term is formed by multiplying the preceding term by the common multiplier, 2.

674. A geometrical progression is an ascending or descending series, according as the common multiplier is a whole number or a fraction. Thus, the ascending series, 1, 2, 4, 8, 16, etc., is a geometrical progression in which the common multiplier is 2; and the descending series, 32, 16, 8, 4, 2, 1, 1, 1, etc., is a geometrical progression in which the common multiplier is.

675. The Ratio in a geometrical progression is the common multiplier.

676. In the solution of problems in Arithmetical or Geometrical progression, five parts or elements are concerned, viz:

The conditions of a problem in progression may be such as to require any one of the five parts from any three of the four remaining parts; hence, in either Arithmetical or Geometrical Progression, there are 5 × 4 = 20 cases, or classes of problems, and no more, requiring each a different solution.

V

GENERAL PROBLEMS IN ARITHMETICAL PROGRESSION.

PROBLEM I.

677. Given, one of the extremes, the common difference, and the number of terms, to find the other

extreme.

Let 2 be the first term of an arithmetical progression, and 3 the common difference; then,

From this illustration we perceive that, in an arithmetical progression, when the series is ascending, the second term is equal to the first term plus the common difference; the third term is equal to the first term plus 2 times the common difference; the fourth term is equal to the first term plus 3 times the common difference; and so on. In a descending series, the second term is equal to the first term minus the common difference; the third term is equal to the first minus 2 times the common difference; and so on. In all cases the difference between the two extremes is equal to the product of the common difference by the number of terms less 1. Hence the

RULE. Multiply the common difference by the number of terms less 1; add the product to the given term if it be the less extreme, and subtract the product from the given term if it be the greater

extreme.

EXAMPLES FOR PRACTICE.

1. The first term of an arithmetical progression is 5, the common difference 4, and the number of terms 8; what is the last term? Ans. 33.

2. If the first term of an ascending series be 2, and the common difference 3, what is the 50th term?

3. The first term of a descending series is 100, the common difference 7, and the number of terms 13; what is the last term? 4. If the first term of an ascending series be, the common difference, and the number of terms 20, what is the last term?

Ans. 72.

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