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3. What is the cube root of 21952 ?

Ans. 28. 4. What is the cube root of 250047 ?

Ans. 63. 5. What is the cube root of 614125 ?

Ans. 85. 6. What is the cube root of 21024576 ? Ans. 276. 7. What is the cube root of 94818,816 ? Ans. 45,6. 8. What is the cube root of 7612,812161 ? Ans. 19,67 + 9. What is the cube root of ,121861281 ? Ans. ,495 x 10. What is the cube root of ,000021952 ? Ans. ,028. 11. What is the cube root of ?

Thus, 8=2, the numerator, and ✓ 27=3, the denominator.

Ans. 12. What is the cube root of 313

Ans. 13. What is the cube root of 1?

Ans. 14. What is the cube root of 5 12 ?

Ans. 15. What is the cube root of 3197

8 9

Ans. 13

4096

16.

APPLICATION OF THE CUBE ROOT.

1. How many solid or cubic feet are contained in a cubical block which is 8 feet long, 6 feet wide and 4 feel thick?

8 X 6X4=192, Ans. 2. How many solid or cubic feet of earth were thrown out of a cellar which is 32 feet long, 25 feet wide, and 10 feet deep?

Ans. 8000. 3. What is the length of the side of a cube which con. tains 8000 solid or cubic feet? * 8000=20ft. Ans.

4. A bushel contains 2150,420 solid or cubic inches; what is the length of the side of a cubic box which shall contain that quantity ?

Ans. 12,9+inches. 5. The side of a certain cubical box measures 1 foot

i what is the length of the side of another that is 8 times as large ?

1x1x1=1x8=8, and Ý 8=2 feet, Ans.

examine our cube, with all these additions made and placed to it, we shall discover in one comer a vacancy, the length, breadth and thickness, of which is just 3 inches ; (that is, the same as the thickness of our last addition ; which, when filled, will just complete the cube. This vacancy contains 3X3X3=27 cubic inches ; that is, the cube of the last quotient figure. These 27 cubic inches we place under the former products, then add them up, and subtract the amount from the dividend, 4167, and 0 remains. Hence we find, that the side of a cube which contains 12167 inches must measure 23 inches; or, that the cube root of 12167 is 23. Proof.-23, that is, 23 X 23 X 23=12167, the given sum, therefore right.

Ans. 6416

Note. The solid contents of similar figures are in proportion to each other as the cubes of their similar sides or diameters.

6. If a ball 4 inches in diameter weigh 12 pounds, what will another ball of the same metal weigh, whose diameter is 7 inches? 4X4X4= 64 in. lbs. in.

lbs. Then, 7x7x7=343

64 : 12 :: 343 : 7. If a globe of silver 3 inches in diameter be worth 150 dollars what is the value of a globe 8 inches in diameter ?

Ans. $2844,44+ 3. dow many globes 1 foot in diameter would it take to make a globe 2 feet in diameter ?

Ans. 8. 9. The diameter of a ball weighing 4 pounds, is three inches ; what is the diameter of another ball 8 times as large?

3x3x3=27, and 27 X 8= 216=6 inches, Ans. 10. If the side of a cube of silver worth 20 dollars, be 2 inches, what is the side of another cube of silver, whose value shall be 64 times as much?

Ans. 8 inches. 11. If the diameter of the earth is 8000 miles, and the sun is one million times as large as the earth, what is the diameter of the sun ? Ans. Eight hundred thousand miles,

Prob. 1.—The product of two or more parts of any num. ber given, to find that number.

RULE.

Divide the given product by the product of the given parts, and the quotient will be that power of the required number which is equal to the number of parts.

Ex. 1. If į and of a certain number be multiplied together the product will be 54 ; what is that number?

Thus, įx= , then 54-3=144, which is the 2d pow. er of the required number, because the number of parts multiplied were 2; then v144=12, Ans.

Ex. 2. If %, and 5 of a certain number be multiplied together the product will be 12000 : what is that number?

Thus, xxã==; then 12000:=27000, which is the 3d power of the required number; and 27000= 30, the Answer.

PROB. 2. The product of any two or more numbers, and the proportion between them given, to find those numbers.

RULE.

Divide the given product by the product of the given terms of the proportion, and the quotient will be a power equal to the number of terms multiplied for a divisor; and the root of that power, multiplied severally by the given terms of the proportion, will produce the required numbers.

Èx. 1. The product of two numbers is 2240, and they are in proportion to each other as 5 to 7; what are those numbers ?

5x7=35)2240(64, and 64=8; Then 8x5, (one of the terms of the propor. tion,) gives

40

Ans. And 8 x7, (the other term of the proportion,) gives

56 Ex. 2. The product of three numbers is 1296, and they are in proportion to each other as 1, 2 and 3 ; what are those numbers ? Thus, 1x2x3=6)1296(216,

1= 6

2=12 Ans then 216=6

3=18

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A GENERAL RULE FOR EXTRACTING ROOTS OF ALL

POWERS.

RULE.

1. Prepare the given number for extraction, by pointing it off from the units' place, as the required root directs.

2. Find the first figure of the root by trial, and subtract its power from the left hand period.

3. Bring down the first figure in the next period to the remainder and call this the dividend.

4. Involve the root to the next inferior power to that which is given, and multiply it by the number denoting the given power for a divisor.

5. Find how many times the divisor may be had in the dividend, and the quotient will be another figure of the root.

6. Involve the whole root to the given power, and subtract that power from as many periods of the given number,

have found figures in the root.

as you

7. Bring down the first figure of the next period to the remainder for a new dividend.

8. Involve the whole root to the next inferior power to that which is given, and multiply it by the number denoting the given power for a divisor, as before ; and proceed in this manner till the whole is finished.

Note. When the number to be subtracted is greater than the periods from which it is to be subtracted, the last quotient figure must be taken less, &c.

EXAMPLES.

1. What is the cube root of 94818,816 ?

94818,816(45,6 root.

64
4x4x3=48)308 dividend.

45 X 45 X 45=91125 subtrahend.
45 X 45 X3=6075)3693,8 dividend.
456 x 456 X 456=94818,816 subtrahend.

00000,000 2. What is the sursolid, or 5th root of 17210368 ?

17210368(28 root.

32 2x2x2x2=16X4=64)1401 dividend. 28 X 28 X 28 X 28 X 28=17210368 subtrahend.

00000000 Note. The roots of most powers may be found by the square and cube roots only, by the following

RULE.

1. For the biquadrate, or 4th root, extract the square root of the square root.

2. For the 6th root, extract the cube root of the square root.

3. For the 8th root, extract the square root, which reduces it to the 4th power; then extract the root of that power as above.

4. For the 9th root, extract the cube root of the cube root.

5. For the 12th root, extract the square root, which will reduce it to the 6th power; then find the root of the 6th power as above.

EXAMPLES.

1. What is the biquadrate, or 4th root of 20736 Thus, the square root of 20736 is 144 ;

then, the square root of 144 is 12, the Ans. 2. What is the square cubed, or the sixth root of 481890304 ?

Thus, V481890304=21952; and 21952=28, Ans.

3. Extract the square biquadrate, or eighth root of 1001129150390625.

[square

2d power, or Thus, V 1001129150390625=731640625=15625= 75, root, the answer.

4th power.

Questions. 1. What is a cube?

bringing down all the periods, how may 2. What is it to extract the cube root? we continue the operation ?

3. What is a solid body, having six 6. Repeat the rule for extracting the equal sides, each an exact square ? roots of all powers ?

4. What is the Rule for extracting the 7. How may the roots of most powers cube root?

be found ? 5. When there is a remainder after

ALLIGATION. Alligation is the method of mixing several simples of different qualities, so that the composition may be a mean, or middle quality. It consists of two kinds, Alligation Medial and Alligation Alternate.

ALLIGATION MEDIAL, Is when the quantities and prices of several things are given, to find the mean price of the mixture composed of those materials.

RULE.

As the sum of the quantities, or whole composition, is to the whole value, so is any part of the composition, to its mean price, or value.

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