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INVOLUTION,

OR THE RAISING OF POWERS.

A POWER is the product arising from multiplying any given number into itself continually a certain number of times, thus:

3x3=9-32, the 2d power or square of 3.

3×3×3=27=33, the 3d power or cube of 3.

3x3x3x3=81, the 4th power 34 or the biquadrate of

3, &c.

The number denoting the power is called the index of the power.

RULE.

Multiply the number, root, or first power, continually by itself, till the number of multiplications be one less than the index of the power to be found, and the last product will be the power required.

NOTE.-Vulgar fractions may be involved to any power by raising each of the terms (numerator and denominator) to the power required; and if a mixed number be proposed, reduce it to an improper fraction, and then proceed as above directed. Or,

A vulgar fraction may be reduced to a decimal, and then involved as a whole number.

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degree of accuracy. The roots to such numbers, which are perfectly found, are called rational roots.

The roots which approximate, that is, those that cannot be accurately expressed in numbers, are called surd roots: thus the square root of 5 is a surd root.

The sign ✔prefixed to a number, denotes its root; thus, ✓5 denotes the square root of 5.

3/8 indicates the cube root of 8.

16 denotes the 4th root of 16, &c.

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If the root be expressed by several numbers with the sign+or between them, a line is drawn over all the parts of it: thus, the square root of √25+11=6; and the cube root of 820-91-9.

EXTRACTION OF THE SQUARE ROOT.

RULE.

1. Separate the given number into periods of two figures each, beginning at the unit figure, or decimal point.

2. Find the greatest square in the first (left nand) period, and set its root on the right hand of the given number, after the manner of the quotient figure in Division.

3. Subtract the square thus found, from the said period, and to the remainder, annex the second period, for a dividend. 4. Double the root already found, for a divisor, and find how often it is contained in the dividend, exclusive of the place of units; annex the result* to the quotient and also to the divisor.

5. Then multiply and subtract as in Division, and to the remainder, annex the third period for a new dividend.

6. To the (last) divisor, add the last figure of the root for a new divisor, find another quotient figure as before, annex it to the root, and the divisor, multiply and subtract as before.

NOTE 1.-If there be decimals in the given number, it must be pointed both ways from the unit or decimal point, and the root will consist of as many figures of whole numbers and decimals respectively as there are periods of integers or decimals in the power. NOTE 2.-The root of a vulgar fraction is found by reducing it

*The quotient found by this mode is sometimes too great: when this is the case, it must be diminished until the complete divisor, multiplied by the quotient figure, produces a result equal to or less ban the dividend.

to its lowest terms, and extracting the root of the numerator for a new numerator, and of the denominator for a new denominator.

PROOF.

Square the root found, adding in the remainder, if any.

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2. Required the square root of 106929.

Ans. 327.

3. Required the square root of 152399025. Ans. 12345. 4. Required the square root of 119550669121.

5. Required the square root of 368863.

Ans. 345761.

Ans. 607.34092+

6. Required the square root of 3.1721812.

Ans. 1.78106+

7. How much is the square root of .00032754?

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Ans. .01809+
Ans..

EXTRACTION OF THE CUBE ROOT.

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A cube is any number multiplied by its square. tract the cube root is to find a number, which, being multiplied into itself and then into that product, will produce the given number.

RULE.

1. Distinguish the given number into periods of three figures each, beginning at the unit's place, completing the periods of decimals with ciphers, when necessary.

2. Find the greatest cube in the left hand period, and subtract it therefrom, and put its root in the quotient, and to the remainder annex the next period for a dividend.

3. Square the quotient, and multiply it by 300 for a trial divisor; find how often it is contained in the dividend, and put the result in the quotient.

4. Multiply the figure last put in the quotient, by the rest, and the product by 30; and to the product add the square of the last figure put in the quotient, for the second part of the divisor, and the sum of these complete the divisor.

5. Multiply the complete divisor by the figure last put in the quotient, subtract the product from the dividend, bring down the next period, and proceed as before.

NOTE.-The precepts under the square root are equally applicable here.

PROOF.

Involve the root found to the third power, adding in the remainder, if any; the result will be equal to the given number.

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Ans. 325.

Ans. 327.

Ans. 4.39

Ans. 618.

Ans

2. Required the cube root of 34328125. 3. Required the cube root of 34965783. 4. Required the cube root of 84.604519 5. Required the cube root of 236029032. 6. How much is the cube root of 729 ?

1331

A GENERAL RULE FOR EXTRACTING THE ROOTS OF ALL POWERS.

1. Prepare the given number for extraction, by pointing off from the units, as the required root denotes; thus, for the square root two figures, cube root three, the fourth root four, &c.

*24300 the first trial divisor, and 2881200 the second do. do.

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