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3. Required the square root of 2 to 12 places.

2(1.41421356237 +- root.

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8 4. What is the square root of 152399025 ?

Ans. 12345 5. What is the square root of "00032754?

Ans. '01809. 6. What is the square root of I?

6. What

Ans. .645497

7. What is the square root of o?

Ans. 2'5298, &c.

8. What is the square root of 10?

Ans. 3:162277, &c.

TO EXTRACT THE CUBE Roor,

RULE.*

1. Having divided the given number into periods of 3 figures, find the nearest less cube to the first period by the table of powers or trial ; eet its root in the quotient, and subtract the said cube from the first period ; to the remainder bring down the second period, and call this the resolvend.

2. To three times the square of the root, just found, add three times the root itself, setting this one place more to the right than the former, and call this sum the divisor. Then divide the resolvend, wanting the last figure, by the divisor, for the next figure of the root, which annex to the former ; calling this last figure e, and the part of the root before found call a.

3. Ada

* The reason of pointing the given number, as directed in the rule, is obvious from Cor. 2, to the Lemma made use of in demonstrating the square rout ; and the rest of the operation will be best understood from the following analytical process :

Suppose N, the given number, to consist of two periods, and let the figures in the root be denoted by a and b.

Thes

3. Add together these three products, namely, thrice the square of a multiplied by e, thrice a multiplied by the square of e, and the cube of e, setting each of them one place more to the right hand than the former, and call the sum the subtrahend ; which must not exceed the resolvend ; and if it do, then make the last figure « less, and repeat the operation for finding the subtrahend.

4. From the resolvend take the subtrahend, and to the remainder join the next period of the given number for a new resolvend ; to which form a new divisor from the whole root now found ; and thence another figure of the toot, as before, &c.

EXAMPLES.

Then a +51'=a} +3a*b* 3aba +53=N= given number, and to find the cube root of N is the same as to find the cube root of al +3a2b+zaba + b3 ; the method of doing which is as follows:

a +3a*b+ 3ab:+63 ( atbroot.

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And in the same manner may the root of a quantity, consisting of any number of periods whatever, be found.

EXAMPLES.

1. To extract the cube root of 48228-544. 3X3 27

48228.544(36-4 root. 3X3 09 | 27

Divisor - 279

279 21228 resolvend.

[blocks in formation]

Ans. 103.

2. What is the cube root of 1092727 ?
3. What is the cube root of 27054036608 ?

Ans. 3002.

4. What is the cube root of .0001357 ?

Ans. '05138, &c. 5. What is the cube root of 15.30

Ans. 6. What is the cube root of

?

Ans. •873, &c.

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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

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

Let

Let N= given number, a' = supposed cube, and x cortection.

Then 2a? +N : 2Nta? :: 0 : at-x by the rule, and con

sequently zal + Nxat-x=2N fai xa; or 2a' tatx'Xa+*

=2Nta' xa.

Or 224 +2a’x+44 +425x + 60*20* + 40x3 + x4=2aNta", and by transposing the terms, and dividing by 22

N=a'+32*x+3x2+x**** ****which by neglecting the

2a

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as being very small, becomes N=a' + 3a'xt

2a

30x+*= the known cube of a+x. Q. E. I.

W.

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