THE CUBE ROOT.

The cube root of a given number, is such a number as being multiplied by itself, and then into that product, produces the given number.

RULE.

1. Point off the sum into periods of three figures each, beginning with units.

2. Find the greatest cube in the left hand period, place the root of it in the quotient, subtract the cube from the left hand period, and to the remainder bring down the next period for a resolvend.

3. Square the quotient and multiply the square by 3 for a defective divisor.

4. Seek how often the defective divisor is contained in the resolvend, omitting the units and tens, or two right hand figures. Place the result in the quotient, and its square to the right of the divisor, supplying the place of tens with a cypher, whenever the square is less than ten.

5. Multiply the last figure of the quotient or root by all the figures in it previously ascertained; multiply that product by 30, and add their product to the divisor, to complete it.

6. Multiply and subtract as in Simple Division, and to the remainder bring down the next period, for a new resolvend. Find a divisor as before, and thus proceed until all the periods are brought down.

Note.-The cube root of a vulgar fraction is found by reducing it to its lowest terms, and extracting, as in the square root; and if the fraction be a surd, reduce it to a decimal, and then extract the root.

In extracting the cube root, if the sum be in part decimals, or if the whole be decimals, point the figures as in the square root, observing to have three figures in a period instead of two; and in all cases in the cule root, when there is a remainder, if it be required to obtain decimal figures to the root, proceed as directed in the square root, only add three cyphers, in place of two to the remainder.

PROOF.

Involve the root to the third power, adding the remain der, (if any,) to the result...

2. What is the cube root of 48228.544? Ans. 36.4.

3. What is the cube root (or 3d root) of 2?

Ans. 1.259921.

4. What is the cube root of 132651?
5. What is the cube root of 4173281?
6. What is the cube root of .008649? Ans
7. What is the cube root of 15?

Ans. 51. Ans. 161.

2052+. Ans. .

8 What will be the cube root of 160, the decimal being continued to three places?

Ans. 5.4284.

9. If the contents of a globe amount to 5832 cubick inches, what are the dimensions of the side of a cubick block containing the same quantity? Ans. 18 in. square.

10. If the diameter of the planet Jupiter is 12 times as much as the diameter of the earth, how many globes of the earth would it take to make one as large as Jupiter? Ans. 1728.

11. If the sun is 1000000 times as large as the earth, and the earth is 8000 miles in diameter, what is the diameter of the sun? Ans. 800000 miles.

Note. The roots of most powers may be found by the square and cube roots only; thus the square root of the square root is the biquadrate, or 4th root, and the sixth root is the cube of this square root.

Questions concerning the powers and roots.

1. What is called a power?

2. What power is the square?

Ans. The 2d

power.

3. What is the cube of a number called?

4. How do you raise the power of a vulgar fraction? 5. What is the root of a power?

6. What is meant by extracting the square root? 7. Repeat the rule for doing it.

8. How do you proceed when the sum consists in part, or altogether, of decimals?

9. How do you extract the root of a vulgar fraction? 10. How do you proceed when the fraction is a surd? 11. What do you understand by the cube root? 12. Repeat the rule for extracting it.

13. How are sums in the square root proved? 14. How are sums in the cube root proved?

ALLIGATION.

Alligation is a rule for mixing simples of different qualities, in such a manner that the composition may be of a mean or middle quality.

CASE I.

To find the mean price of any part of the mixture, when the quantities and prices of several things are given.

RULE.

As the sum of the quantities is to any part of the composition, so is the price of the quantities to the price of any particular part.

EXAMPLES.

1. A trader mixes 60 gallons of wine at 100 cents per gallon; 40 gallons, at 80 cents, and 30 gallons of water. What should be the price per gallon?

2. A trader mixes a quantity of tea as follows, viz:4 lbs. of tea at 42 cents per lb.; 6 lbs. at 33 cents; 12 lbs. at 75 cents, and 15 lbs, at 80 cents. What can he sell it for per lb.? Aus. 6634 cents.

3. A farmer mixes 20 bushels of wheat at 5s. per bushel, with 383 bushels of rye at 3s, and 40 bushels of arley at 2s. per bushel; how much is a bushel of the mixture worth? Ans. 3s.

CASE II.

When the prices of several simples are given to find what quantity of each, at their respective prices, must be taken to make a compound at a proposed price.

RULE.

Set the prices of the simples in a column under each other. Connect with a continued line, the rate of each simple which is less than that of the compound, with one or any number of those that are greater than the compound, and each greater rate, with one or more of the ess. Place the difference between the mixture rate, and that of each of the simples, opposite to the rates, with which they are linked. Then, if only one difference stand against any rate, it will be the quantity beonging to that rate; but if there be several, their sum will be the quantity. Different modes of linking, will produce different answers.

EXAMPLES.

1. A merchant would mix wines at 17s. 18s, and 22s. per gallon, so that the mixture may be worth 20s. per gallon: what quantity of each must be taken?

2 at 17s.

2 at 18s.
3+2=5 at 22s.

Ans. 2 gallons at 17s., 2 at 18s., and 5 at 22s.

2. How much barley at 40 cents, corn at 60, and wheat at 80 cents per bushel, must be mixed together, that the compound may be worth 62 cents per bushel? Ans. 17 bush. of barley, 174 of corn, and 25 of wheat.

CASE III.

When the price of all the simples, the quantity of one of them, and the mean price of the mixture, are given, to find the quantities of the other simples.

RULE.

Find an answer as before, by connecting; then, as the difference of the same denomination with the given quanity, is to the differences respectively, so is the given quantity, to the different quantities required.

EXAMPLES.

1. How much gold of 15, 17, 18, and 22 carats fine must be mixed together to form a composition of 40 oz. of 20 carats fine?

16

Then as 16 : 2 : : 40: 5

and as 18: 10:40:25 Answer.

Ans. 5 oz. of 15, 17 and 18 carats fine, and 25 oz, of 22 carats fine.

2. A grocer has currents at 4d., 6d., 9d., and 11d., per b. and he would make a mixture of 240 lbs, that migh: be sold at 8d. per lb.; how much of each kind must he take?

Ans. 72 lbs. at 4d., 24 at 6d., 48 at 9d, and 96 at 11d.

CASE IV.

When the prices of the simples, the quantity to be mixed, and the mean price are given, to find the quantity of each simple.

RULE.

Connect the several prices, and place their differences as before; then, as the sum of the differences thus given, is to the difference of each rate, so is the quantity to be compounded, to the quantity required.