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ceed? Repeat the rule. When the given number is a whole number, how many figures will there be in the root? When you can not get the exact root, what can be done? How can you extract the root of a decimal fraction? What will you do with a whole number and decimal? What is the rule? How will you extract the square root of a vulgar fraction? Repeat the rule? Give an example on the slate. What is the difference between a square and the square root?

47. Required the square root of 9876.4792144? (This is given for a trial question.)

CUBE ROOT.

THE Cube of a number is the product of that number multiplied into its square. To extract the cube root, is to find a number which being multiplied into itself, and then into that product, will produce the given number.

ILLUSTRATION.

The solid called a cube, has its length, breadth, and thickness, all equal. As the number of solid feet, inches, &c., in a cube, are found by multiplying the length, height, and breadth, together-that is, by multiplying one side into itself twice, the third power of a number is called the cube of that number; thus a cubical or solid foot has 6 equal sides of 1 foot square, consequently it follows that a cubical foot contains 1728 solid or cubical inches, because 12 × 12=144 × 12=1728 inches: that is, a cubical foot or block of wood may be sawn into 1728 blocks, each of which will be a solid inch, which may be represented in the following manner :

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Figure 2 is a solid or cubical foot with 6 equal sides; suppose this block be sawn into 12 pieces at the distance of 1 inch, it would make 12 pieces, as represented above, each 1 foot square, and 1 inch in thickness, as in figure 1 (S. R.); now, each of those 12 square pieces may be divided into 144 solid blocks the 12 blocks divided would be 12 × 12=144 inches for each of the 12 pieces, as represented by the 12 squares, which is equal to 1728 inches in a cubical or solid foot, and 12 is the cube root of 1728.

Thus, the root 12 is equal to 12 square feet of 1 inch in thickness (as 1, 2, 3, &c.), or 12 inches in length, breadth, and thickness, which is equal to 1 solid foot.

Thus, 1×300=300: 2×30=60 : 2×2=-4

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In the example above, I first point off the three figures from the right, and place the period over the 7; this leaves 1, or unity, for the first operation; consequently the quotient figure must in this case be 1, which I put in the quotient, and under 1 in the dividend; then bring down the next period (728); now seek for the divisor; first multiply 1 by 300-300 for a trial divisor, then suppose 300 in 728, say 2, which put in the quotient; then 2 x 30-60; then 2×24 the square of 2; then add these several products together and you have the true divisor (364) which multiply by the last quotient figure (2) and the work is finished.

RULE I.

1. Point the given number into periods of three figures each, beginning at the right or units' place.

2. Find the greatest cube in the left-hand period, and subtract it therefrom, and set down the root in the quotient, and then to this remainder bring down and annex the next period for a dividend.

3. Square the quotient by multiplying it into itself, and multiply that product by 300 for a trial divisor; see how often it is contained in the dividend, and set the result in the quotient.

4. Multiply the figure last put in the quotient by the other figures in the quotient, and that product by 30.

5. Then square this last figure in the quotient, and add it to the product just mentioned, for the second part of the divisor; all these products added, form the true divisor.

6. Now multiply this true divisor by the last figure put in the quotient; subtract the product from the dividend as in division, bring down the next period of three figures; proceed as before.

For Decimals.-Annex ciphers to the decimal, if necessary, so that it shall consist of 3, 6, 9, &c., places; then put the first point over the place of thousandths, the second over the place of millionths, and so on over every third place to the right; and then extract the root as in whole numbers. Observe, 1. There will be as many places in the root as there are periods in the given number. 2. The same rule applies when the given number is composed of a whole number and a decimal. 3. If there be a remainder in a whole number, after all the periods have been brought down, you can annex periods of ciphers, considering them as decimals.

For Vulgar Fractions.-1. Reduce compound fractions to simple ones, mixed numbers to improper fractions, and then reduce the fraction to its lowest terms. 2. Then extract the cube root of the numerator and denominator separately, if they have exact roots; but if either of them have not an exact root, reduce the fraction to a decimal, and extract the root as above. Proof-Cube the root and add in the remainder.

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1. Point off the given number into periods of three figures each, if whole numbers commencing at the right hand; but if decimals, at the left.

2. Find the greatest cube in the left hand period, and place the root to the right of the given number, and subtract the cube of the root from the left hand period, and to the remainder bring down the next period for a dividend.

3. Square the root, and multiply it by three for a defective divisor.

4. Reserve mentally the units and tens of the dividend, and try how often the defective divisor is contained in the remainder; place this result to the root, and the square of it to the right of the defective divisor, but if the square is less than ten, supply the ten's place by a cipher.

5. Complete the divisor by adding thereto the product of the last figure of the root, and the remaining figure or figures of the root, and that again by 30; then divide and subtract as in long division.

6. The defective divisors after the first may be easily found by adding to the last divisor the number that completed it, with twice the square of the last figure of the root.

7. The divisor is then completed according to the 4th and 5th paragraphs above.

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3. What is the cube root of 99252.847? 4. What is the cube root of 259694072 ? 5. What is the cube root of 34328125 ? 6. What is the cube root of 12.977875 ?

7. What is the cube root of 171.46776406 ?

8 What is the cube root of .5?

Ans. 638.

325.

2.35.

5.555+

17+87 rem

9. What is the cube root of 32461759 ? 10. What is the cube root of 27054036008 ? 11. What is the cube root of .751089429 ? 12. What is the cube root of 3.408862625 ? 13. What is the cube root of 250 ? 14. What is the cube root of 31.15?

686

343

15. What is the cube root of ?
5
16. What is the cube root of 4913 ?

9261

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26. What is the cube root of 436036424287 ? 27. What is the cube root of 99?

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APPLICATION.

To find two mean proportionals between any two given numbers

RULE.

Divide the greater by the less, and extract the cube root of the quotient.

2. Multiply the root so found by the least of the given numbers, and the product will be the least.

3. Multiply this product by the same root, and it will give the greatest.

28. What are the two mean proportionals between 6 and 750?

Thus: 750-6-125 and 3/125=5, then 5×6=30, least, and 30×5=150 greatest; 30 and 150. Ans.

NOTE. The solid contents of similar figures are in proportion to each other, as the cubes of their similar sides or diameters. 29. If a bullet 6 inches in diameter weigh 32 lb., what will a buliet of the same metal weigh, whose diameter is 3 inches? As 6×6×6=216; 3x3x3=27; as 216: 32 lb. :: 27: 4 lb.

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