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CHAPTER XI. .

INVOLUTION. 175. INVOLUTION is the method of finding the powers of numbers.

We have already defined the power of a number to be the result arising from multiplying the same into itself continually, until the number has been used as a factor as many times as there are units in the exponent denoting the power. Thus, to obtain the cube, or third power of 7, we must use it as a factor three times, which will produce 7x7x7=343. . . .

Examples. 1. What is the square of 23? .

Ans. 529. 2. What is the cube of 17 ?

· Ans. 4913. 3. What is the 5th power of 47 ?

Ans. 229345007. 4. What is the 9th power of 9 ? :

Ans. 387420489 5. What is the square of 22667121?

Ans. 513798374428641. 6. What is the square of 625?'

Ans. 390625. 17. What is the cube of 48?

Ans, 110592. 8. What is the cube of 681472?

Ans. 316478381828866048. 9. What is the square of 0.75 ?

Ans. 0.56252 10. What is the cube of 0.65?

Ans. 0.274625. 11. What is the square of 81 ?

Ans. 727.

EVOLUTION. 76. EVOLUTION is the reverse of involution. It explains the method of resolving a number into equal factors, which factors are called roots.

When the number is resolved into two equal factors, this factor is called the square root of the number:

When a number is resolved into three equal factors, the factor is called the cube root of the number.

The operation of resolving a number into two equal factors is called the extraction of the square root.

EXTRACTION OF THE SQUARE ROOT. 77. If we square 48 by the usual rule, we get 48° =2304: But if instead of 48, we use 40+8, we shall find, by actual multiplication,

40+8

40+8 1600+320

.+320+64

1600+ 640+64 for the square of 40+8=48. Now to reverse this operation, that is, to extract the square root of 1600+640+64, we proceed as follows :

We take the square root of 1600, which is 40; this is the first part of the root; its square being subtracted from 1600+640+ 64, leaves the remainder 640+ 64. We see that 640, divided by twice 40, or 80, gives 8 for a quotient, which is the second part of the root required.

CASE I. From the above process, we deduce the following rule for the extraction of the square root of a whole number.

RULE.

1. Point off the given number into periods of two figures each, counting from the right towards the left. When the number of figures is odd, it is evident that the left-hand, or first period, will consist of but one figure.

II. Find the greatest square in the first period, and place its root at the right of the number, in the form of a quotient figure in division. Subtract the square of this root from the first pe. riod, and to the remainder annex the second period; the result will be the FIRST DIVIDEND.

III. Double the root already found, and place it on the left of the number, for the FIRST TRIAL DIVISOR. See how many times this trial divisor, with a cipher annexed, is contained in the dividend, the quotient figure will be the second figure of the root; this must be placed at the right of the TRIAL DIVISOR; the result will be the TRUE DIVISOR. Multiply the true divisor by this second figure of the root, and subtract the product from the dividend, and to the remainder annex the next period, for a BECOND DIVIDEND.

IV. To the last TRUE DIVISOR, add the last figure of the root, for a new TRIAL DIVISOR, and continue to operate as before, until all the periods have been brought down..

Examples. 1. What is the square root of 531441 ?

First trial divisor, 14 . first true divisor, 142 second trial divisor, 144. second true divisor, 1449

53'14'41 (729 root
49
414 first dividend,
284
13041 second dividend,
13041

0 .

2. What is the square root of 11390625?

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In the first example, we exhibited the trial divisors, as well as the true divisors, but in the second example we adhered more closely to our rule, and placed the succeeding figures of the root at the right of the trial divisors, without again wri. ting them down.

3. What is the square root of 11019960576?

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4. What is the square root of 16983563041 ? .

Ans. 130321. 5. What is the square root of 79792266297612001?

Ans. 282475249. 6. What is the square root of 852891037441 ?

Ans. 923521. 7. What is the square root of 61917364224?

Ans. 248832. 8. What is the square root of 13422659310152401?

Ans. 115856201.

CASE II. To extract the square root of a decimal fraction, or of a number consisting partly of a whole number, and partly of a decimal value, we have this

RULE. · 1. Annex one cipher, if necessary, so that the number of decimals shall be even.

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