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INVOLUTION.

639. A Power is the product arising from multiplying a number by itself, or repeating it any number of times as a factor. 640. Involution is the process of raising a number to a given

power.

641. The Square of a number is its second power.

642. The Cube of a number is its third power.

643. In the process of involution, we observe,

I. That the exponent of any power is equal to the number of times the root has been taken as a factor in continued multiplication. Hence

II. The product of any two or more powers of the same number is the power denoted by the sum of their exponents, and III. If any power of a number be raised to any given power, the result will be that power of the number denoted by the product of the exponents.

1. What is the 5th power of 6?

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tiplication; the final product, 7776, is the power required, (I). Or,

we may first form the 2d and 3d powers then the product of these two powers will be the 5th power required, (II).

2. What is the 6th power of 12?

OPERATION.

122 = 144

1443 2985984, Ans.

=

ANALYSIS. We find the cube of

the second power, which must be the 6th power, (III).

644. Hence for the involution of numbers we have the fol

RULE. I Multiply the given number by itself in continued multiplication, till it has been takeu as many times as a factor as there are umts in the exponent of the required power. Or,

II. Multiply together two or more powers of the given number, the sum of whose exponents is equal to the exponent of the required power. Or,

III Raise some power of the given number to such a power that the product of the two exponents shall be equal to the exponent of the required power.

NOTES.

1. A fraction is involved to any power by involving each of its terms separately to the required power.

2 Mixed numbers should be reduced to improper fractions before involution. 3. When the number to be involved is a decimal, contracted multiplication may be applied with great advantage.

EXAMPLES FOR PRACTICE.

1. What is the square of 79?
2. What is the cube of 25.4?
3. What is the square of 1450?
4. Raise 164 to the 4th power.
5. Raise 2 to the 20th power.

Ans 6241. Ans. 16387.064.

Ans. 796592

Ans. 1048576

6. Raise .4378565 to the 8th power, reserving 5 decimals

Ans. .00135 ±

7. Raise 1.052578 to the 6th power, reserving 4 decimals

8. Involve .029 to the 5th power?

Ans. 1.3600 ±

Ans. .000000020511149.

Ans. 363.691178934721.

Find the value of each of the following expressions:

9. 4.367*

10 (7)3.

11. (23)5

12 4.63 x 253

13. (63)-7.25'.

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14. (84) × 2.51

Ans. 5.

Ans. 1200

EVOLUTION.

645. A Root is a factor repeated to produce a power; thus, in the expression 7 × 7 × 7 343, 7 is the root from which the power, 343, is produced.

=

646. Evolution is the process of extracting the root of a number considered as a power; it is the reverse of Involution. Any number whatever may be considered a power whose root is to be extracted.

647. A Rational Root is a root that can be exactly obtained. 648. A Surd is an indicated root that can not be exactly obtained.

649. The Radical Sign is the character, ✔, which, placed before a number, indicates that its root is to be extracted.

650. The Index of the root is the figure placed above the radical sign, to denote what root is to be taken. When no index is written, the index, 2, is always understood.

651. The names of roots are derived from the corresponding powers, and are denoted by the indices of the radical sign. Thus, ✔100 denotes the square root of 100, 100 denotes the cube root of 100; 100 denotes the fourth root of 100; etc.

652. Evolution is sometimes denoted by a fractional exponent, the name of the root to be extracted being indicated by the deno语 minator. Thus, the square root of 10 may be written 10 cube root of 10, 10, etc.

; the

653. Fractional exponents are also used to denote both involution and evolution in the same expression, the numerator indicating the power to which the given number is to be raised, and the denominator the root of the power which is to be taken; thus, denotes the cube root of the second power of 7, and is the same as √72; so also 7

3

5

=

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654. In extracting any root of a number, any figure or figures may be regarded as tens of the next inferior order. Thus, in 2546, the 2 may be considered as tens of the 3d order, the 25 as tens of the second order, or the 254 as tens of the first order.

SQUARE ROOT.

655. The Square Root of a number is one of the two equal factors that produce the number. Thus, the square root of 64 is 8, for 8 x 8

64.

To derive the method of extracting the square root of a number, it is necessary to determine

1st. The relative number of places in a number and its square root. 2d. The relations of the figures of the root to the periods of the number.

3d. The law by which the parts of a number are combined in the formation of its square; and

4th. The factors of the combinations.

656. The relative number of places in a given number and its square root is shown in the following illustrations.

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From these examples we perceive

1st. That a root consisting of 1 place may have 1 or 2 places in the

square.

2d. That in all cases the addition of 1 place to the root adds 2 places to the square.

Hence,

I. If we point off a number into two-figure periods, commencing at the right hand, the number of full periods and the left hand full or partial period will indicate the number of places in the square root.

To ascertain the relations of the several figures of the root to the periods of the number, observe that if any number, as 2345, be decomposed at pleasure, the squares of the left hand parts will be re lated in local value as follows:

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II. The square of the first figure of the root is contained wholly in the first period of the power; the square of the first two figures

of the root is contained wholly in the first two periods of the

and so on.

power;

NOTE. The periods and figures of the root are counted from the left hand. The combinations in the formation of a square may be shown as follows:

If we take any number consisting of two figures, as 43, and decompose it into two parts, 40+ 3, then the square of the number may be formed by multiplying both parts by each part separately: thus, 40+ 3 40 + 3

120 +9

1600 + 120

432 = 1600+240 + 9 = 1849.

Of these combinations, we observe that the first, 1600, is the square of 40 the second, 240, is twice 40 multiplied by 3; and the third, 9, is the square of 3. Hence,

III. The square of a number composed of tens and units is equal to the square of the tens, plus twice the tens multiplied by the units, plus the square of the units.

By observing the manner in which the square is formed, we perceive that the unit figure must always be contained as a factor in both the second and third parts; these parts taken together, may therefore be factored, thus, 240 +9 (80 + 3) × 3. Hence,

IV. If the square of the tens be subtracted from the entire square, the remainder will be equal to twice the tens plus the units multiplied by the units.

1. What is the square root of 5405778576?

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