by first converting it into a fraction whose denominator is

an exact third power. For example, 3:

approximate root of which is +.

3.53 375

=

the

53

53

But the best mode of approximating the third root either of a whole number or of a fraction, is, to convert it into a fraction, whose denominator is the third power of 10, 100, or 1000, &c.; that is, convert it into 1000ths, 1000000ths, 1000000000ths, &c., and find, the nearest root of the result. The root will then be found in decimals.

17

the root of which is

10

5.103 5000 For example, 5= 103 1000' +17+. If a more accurate root is wanted, we may change 5 to 1000000ths; thus, 5=1888888, the root of which is 171. —— = 1·71

5000000

Nor

Hence, it is evident that the denominator may be omitted, and that it is sufficient to annex three zeros to the number for every additional figure in the root. is it necessary to write all the zeros at once, but we may annex three to the remainder, when an additional figure of the root is required, in the same manner as we bring down successive periods.

In like manner, to find the third root of a vulgar fraction, we change it to a decimal with thrice as many decimal figures as we want decimals in the root.

When the number whose root is sought contains whole numbers and decimals, and the number of decimal figures is not a multiple of three, make it so by annexing zeros; or, commencing at the decimal point, separate the whole numbers into periods by proceeding towards the left, and the decimals by proceeding towards the right, and then complete the right-hand period, if necessary, by annexing zeros.

These preparations being made, the root of a number containing decimals is found in the same way as that of an integral number, care being taken to point off one third as many decimals in the root as there are in the power, including the zeros annexed.

1. Extract the third root of 2, accurate to three decimal figures.

Operation.

2. (1.259+.

1

10'00 (3

6

4

364

2

728

2720'00 (432

180

25

45025

5

225125

468750/00 (46875

3375

81

4721331

9

42491979

4383021.

Extract the third roots of the following numbers, accu

rate to two decimals.

ART. 103.

An equation of the third degree is such as, when reduced to its simplest form, contains at least one term in which there are three, but no term in which there are more than three, unknown factors.

When an equation with one unknown quantity contains the third power only of that quantity, it is called a pure equation of the third degree. tion of this kind.

Thus, x3729 is an equa

1. In a package of cloth there are as many pieces as there are yards in each piece, and it is worth

as many cents per yard as there are yards in a piece. Required the number of pieces and the price per yard, the whole being worth $90.

Let x the number of pieces, also the number of yards

3

in a piece; then == the price per yard in cents. Hence, x2= the whole number of yards; and

z=30 pieces, and there were 30 yards in a piece.

x = 10 cents, price per yard.

2. The length of a rectangular box is twice the breadth, and the depth is of the breadth. The box holds 200 cubic feet. Required the three dimensions.

Remark. The cubical contents of any rectangular space, or rectangular solid, are found by taking the product of the length, breadth, and depth.

3. A pile of wood is 27 feet long, 25 feet wide, and 5 feet high. If the same quantity of wood were in a cubical form, what would be the length of one side of the pile?

4. Two numbers are to each other as 4 to 5, and the sum of their third powers is 5103. Required the numbers.

5. What two numbers are such, that the second power of the greater multiplied by the less makes 75, and the second power of the less multiplied by the greater makes 45?

Let x the greater, and y = the less. Then,

6. The product of two numbers is 28, and 8 times the second power of the greater, divided by the less, gives 98 for a quotient. What are these numbers?

7. The sum of the third powers of two numbers is 2728, and the difference of those powers is 728. Required the numbers.

8. The breadth of a piece of land is of its length, and it is worth as many dollars per square rod, as there are rods in the breadth. The whole piece being worth $9375, what are the dimensions?

9. A gallon being 231 cubic inches, what is the length of one side of a cubical box holding 5 gallons?

10. A bushel being 2150 cubic inches, required the side of a cubical vessel containing 7 bushels.

SECTION XXXIV.

POWERS OF MONOMIALS.

ART. 104. Any power of a quantity is found by multiplying that quantity by itself as many times, less one, as there are units in the exponent of the power. The second power of a or a1 is a. a = a1+1= a2, (Art. 30); this is the same as a1×2.

The third power of a2 is a2. a2. a2=a2+2+2=a¤; this is the same as a2×3.

The second power of a2 b3 is a2 b3. a2b3a2+2b3+3 a4 b6; this is the same as a2× 2 b3×2.

The third power of 3 m2 n3 is 27 m6 n9; this is the same as 27 m2×3 n3 X 3.

Thus we perceive that, in all these examples, we have