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EXAMPLES.

1. To extract the cube root of 48228 544

3X33 27
17 | 48228.544(36.4 root.

3×3 *

09. 27

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2. What is the cube root of 1092727 ?
3. What is the cube root of 27054036008 ?

4.

What is the cube root of 0001357 ?

5130

5. What is the cube root of 1520 6. What is the cube root of

Ans. 103.

Ans. 3002.

Ans. 05138, &c.
Ans.

Ans. 873, &c.

RULE FOR EXTRACTING THE CUBE ROOT BY

APPROXIMATION.*

1. Find by trial a cube near to the given number, and call it the supposed cube.

2. 'Then

* That this rule converges extremely fast may be easily shewa thus:

Let

2. Then, twice the supposed cube added to the given number, is to twice the given number added to the supposed cube, as the root of the supposed cube is to the root required nearly. Or as the first sum is to the difference of the given and supposed cube, so is the supposed root to the difference of the roots nearly.

3. By taking the cube of the root thus found for the supposed cube, and repeating the operation, the root will be had to a still greater degree of exactness.

EXAMPLES.

r. It is required to find the cube root of 98003449.

Let

Let N given number, a supposed cube, and x cor

rection.

Then za3+N: 2N+q3 :: a ax by the rule, and con sequently za3+Nxa+x=2N+a3×a; or 2a3+a+x3×a+x =2N+a3 xa.

Or 2a4+2a3x+aa+4a3x+6a2x2 +44x3 +x^=zaN+aa, and by transposing the terms, and dividing by 20

**

N=a3 +3a*x+3ax2+x3+x3+~~ which by neglecting the

x4

2a

terms x3+, as being very small, becomes Na3 +3a2x+

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2a

34x+x3 the known cube of 4+. Q. E. L

W

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2. Required the cube root of 21035*8.

309

Here we soon find that the root lies between 20 and and then between 27 and 28. Therefore 27 being taken, its cube is 19683 the assumed cube. Then

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1

1242

42

Again for a second operation, the cube of this root is 21035 318645155823, and the process by the latter method is thus:

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As 6310643729 diff. 481355: 276047: the diff.

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

2. Find

* This rule will be sufficiently obvious from the work in the

following example:

Extrac

2. Find the first figure of the root by trial, and subtract its power from the given number.

3. To the remainder bring down the first figure in the next period, and call it the dividend.

4. Involve

Extract the cube root of ao+6a3—40a3 +96a—64. ao+6a5—40a3 +96a—64(a2+2a-4

3a4)6a (+2a

a+6a5 +12a++8a3=a2+za

@2+2a × 3 = 3a + + 12a 3+12a3)—12a+-48a3 +96a—64(—4

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a+6a5-40a3 +96a-64a2+2a-4

When the index of the power, whose root is to be extracted, is a composite number, the following rule will be serviceable :

Take any two or more indices, whose product is the given index, and extract out of the given number a root answering to one of these indices; and then out of this root extract a root answering to another of the indices, and so on to the last.

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Thus, the fourth root square root of the square root,

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The sixth root = square root of the cube root, &c.

The proof of all roots is by involution.

The following theorems may sometimes be found useful in ex

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