GEOMETRICAL EXPLANATION OF CUBE ROOT.

820. What is the length of the edge of a cube whose volume is 15625 cubic feet?

will consist of two figures. The greatest number of tens whose cube is contained in 15000 is 2. Hence, the length of the edge of the cube is 20 feet plus the units' figure of the root. Removing the cube whose edge is 20 feet and whose volume is 8000 cubic feet, there remains a solid whose volume is 7625 cubic feet (Fig. 2). This remainder consists of solids similar to those marked B, C, and D, in Fig. 1 and Fig. 2 of Art. 804.

tion of the 7625 cubic feet, 3 x 202 or 300 × 22 may be used as a trial divisor to find the thickness. Dividing 7625 by 1200 the quotient is 6. But this quotient is too large, for if 6 feet is the thickness, the volume of Fig. 2 will be 3 × 202 × 6+3 x 20 x 62+63, or 9576 cubic feet. Taking 5 feet for the thickness, the volume of Fig. 2 is 7625 cubic feet, for 3x 20 x 5+ 3 x 20 x 52 +5°3 (300 × 22 +30 × 2 ×5+52) 5=1525 x 5=7625. Hence, 25 feet is the length of the edge of a cube whose volume is 15625 cubic feet.

PROBLEMS.

821. 1. What is the length of the edge of a cubical box that contains 46656 cu. inches?

2. What must be the length of the edge of a cubical bin that shall contain the same volume as one that is 16 ft. long, 8 ft. wide, and 4 ft. deep?

3. What are the dimensions of a cube that has the same volume as a box 2 ft. 8 in. long, 2 ft. 3 in. wide, and 1 ft. 4 in. deep?

4. How many square feet in the surface of a cube whose volume is 91125 cubic feet?

5. What is the length of the inner edge of a cubical bin that contains 150 bushels ?

6. What is the depth of a cubical cistern that holds 200 barrels of water?

7. Find the length of a cubical vessel that will hold 4000 gallons of water.

ROOTS OF HIGHER DEGREE.

822. Any root whose index contains no other factors than 2, or 3, may be extracted by means of the square and cube roots.

If any power of a given number is raised to any required power, the result is that power of the given number denoted by the product of the two exponents. (801.) Conversely, if two or more roots of a given number are extracted, successively, the result is that root of the given number denoted by the product of the indices. 1. What is the 6th root of 2176782336 ?

OPERATION.

2176782336 = 46656

46656

= 36

Or,

2176782336 = 1296
1296
= 36

ANALYSIS.-The index of the required root is 62 × 3; hence extract the square root of the given number, and the cube root of this result, which gives 36 as the 6th or required root. Or, first find the cube root of the given number, and then the square root of the result.

RULE.-Separate the index of the required root into its prime factors, and extract successively the roots indicated by the several factors obtained; the final result will be the required root.

2. What is the 4th root of 5636405776?

3. What is the 8th root of 1099511627776?

4. What is the 6th root of 25632972850442049 ?

5. What is the 9th root of 1.577635?

For further practical applications of Involution and Evolution, see "Mensuration."

PROGRESSIONS

823. An Arithmetical Progression is a succession of numbers, each of which is greater or less than the preceding one by a constant difference.

Thus, 5, 7, 9, 11, 13, 15, is an arithmetical progression.

824. The Terms of an arithmetical progression are the numbers of which it consists. The first and last terms are called the Extremes, and the other terms the Means.

825. The Common Difference is the difference between any two consecutive terms of the progression.

826. An Increasing Arithmetical Progression is one in which each term is greater than the preceding one.

Thus, 1, 3, 5, 7, 9, 11, is an increasing progression.

827. A Decreasing Arithmetical Progression is one in which each term is less than the preceding one.

Thus, 15, 13, 11, 9, 7, 5, 3, 1, is a decreasing progression.

828. The following are the quantities considered in arithmetical progression and the abbreviations used for them:

1. The first term, (a). 2. The last term, (7).

3. The common difference, (d). 4. The number of terms, (n). 5. The sum of all the terms, (8).

829. To find one of the extremes, when the other extreme, the common difference, and the number of terms are given.

1. The first term of an increasing progression is 8, the common difference 5, and the number of terms 20; what is the last term?

OPERATION.

20119

19 × 5+ 8 = 103 = 7.

ANALYSIS.-The 2d term is 8+5; the 3d term is 8+ (5 × 2) the 4th term is 8+(5 × 3); and so on. Hence 8+ (19 × 5) or 103 is the 20th or last term. 2. The last term of an increasing progression is 103, the common difference 5, and the number of terms 20; what is the first term?

OPERATION.

20-119

103 19 × 58=a

ANALYSIS.-The 1st term must be a number to which, if 19 × 5 be added, the sum shall be 103; hence, if 19 × 5 is subtracted from 103, the remainder is the first term.

3. The first term of a decreasing progression is 203, the common difference 5, and the number of terms 40; what is the last term?

4. The last term of a decreasing progression is 1, the common difference 2, and the number of terms 9; what is the first term?

RULE.-I. If the given extreme is the less, add to it the product of the common difference by the number of terms less one.

II. If the given extreme is the greater, subtract from it the product of the common difference by the number of terms less one.

FORMULE. = a + (n − 1) × d.

a = 1 - (n − 1) x d.