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EVOLUTION,

OR THE EXTRACTING OF ROOTS.

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The root of a number, or power, is such a number, as being multiplied into itself a certain number of times, will produce that power. Thus 2 is the square root of 4, because 2 X 2 = 4; and 4 is the cube root of 64, because 4 X 4' X 4 = 64, and so on.

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THE SQUARE ROOT. The square of a number is the product arising from that number multiplied into itself.

Extraction of the square root is the finding of such a number as being multiplied by itself will produce the number proposed.

RULE. 1. Separate the given number into periods of two figures each, beginning at the units place.

2. Subtract from the first period the greatest square it contains, setting the root of that square as a quotient figure, and doubling said rcot for a divisor, and bring down the second period to the remainder for a dividual.

3. Try how often the said divisor, (with the figure used in the trial, thereto annexed). is contained in the dividual, and set this figure in both the divisor and root; then multiply and subtract as in division, and bring down the next period.

4. Double the ascertained root for a new divisor, and repeat the

process to the end., Note.-If there are decimals in the given number, point off the periods both ways from the units place; and when the decimals do not consist of an even number of figures, annex a cipher. The root must have as many whole number and decimal figures as there are periods of each in the given number

PROOF. Square the root, adding in the remainder, if any, and the result will equal the given number,

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EXAMPLES.
1. What is the square root of 5499025?

5499025(2345 Ans.
4

2345
43 149

2345 129

11725 464)2090

9380 1856

7035

4690
4685)23425
23425

5499025 Proof. 2. What is the square root of 106929} Ans. 327 3. What is the square root of 451584 ? Ans. 672 4. What is the square root of 363729617 Ans. 6031 5. What is the square root of 7596796 ?

Ans. 2756.228 + 6. What is the square root of 3271.4007?

Ans. 57,19 + 7. What is the square root of 4.372594 ? Ansv 2.091 + 8. What is the square root of 10.4976? Ans, 3.24 9. What is the square root of .00032754?

Ans. .01809 + 10. What is the square root of 10? Ans. 3.1622 + To extract the Square Root of a Vulgar Fraction.

RULE. Reduce the fraction to its lowest terms, then extract the square root of the numerator for a new numerator, and the square root of the denominator for a new denominator.

Note. If the fraction be a surd, that is, one whose root can never be exactly found, reduce it to a decimal, and, extract the root therefrom.

EXAMPLES. 1. What is the square root of 7056?

Ans. [. 2. What is the square root of 1704.

Ans. 3. What is the square root of 75 Ans. .93309 +

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?

478

To extract-the Square Rout of a Mixed Nimber.

RULE. Reduce the mixed number to an improper fraction, and proceed as in the foregoing examples: Or,

Reduce the fractional part to a decimal, annex it to the whole number, and extract the square root therefrom.

EXAMPLES. 1. What is the square root of 3735? Ans. 61 2. What is the square root of 27 16

Ans. 51 3. What is the square root of 851;?

Ans. 9.27 + 4. What is the square root of 35?

Ans. 2.9519 + APPLICATION. 1. A certain square pavement contains 20736 square stones, all of the same size, what number is contained in one of its sides?

Ans. 144.

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2. If 484 trees be planted at an equal distance from each other, so as to form a square orchard, how many will be in a row each way?

Ans. 22. 3. A certain number of men gave 30s. 1d. for a charitable purpose; each man gaye as many pence as there were men : how many men were there? Ans. 19.

Note. The square of the longest side of a right angled triangle is equal to the sum of the squares of the other two sides; and consequently the difference of the square of the longest, and either of the other, is the square of the remaining one.

4. The wall of a certain fortress is 17 feet high, which is surrounded by a ditch 20 feet in breadth; how long must a ladder be to reach from the outside of the ditch to the top of the wall ?

Ans. 26.24 + feet.

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5. A certain castle which is 45 yards high, is surrounded by a ditch 60 yards broad; what length must a ladder be to reach from the outside of the ditch to the top of the castle ?

Ans. 75 yards. 6. A line 27 yards long, will exactly reach from the top of a fort to the opposite bank of a river, which is known to be 23 yards broad; what is the height of the fort?

Ans. 14.142 + yards. 7. Suppose a ladder 40 feet long, be so planted as to reach a window 33 feet from the ground, on one side of the street, and without moving it at the foot, will reach a window on the other side 21 feet high; what is the breadth of the street?

Ans, 56.64 + feet. 8. Two ships depart from the same port; one of them sails due west 50 leagues, the other due south 84 leagues; how far are they asunder? Ans. 97.75+ Or, 973+ lea.

THE CUBE ROOT..

The cube of a number is the product of that number multiplied into its square.

Extraction of the cube root is the finding such a num,ber, as, being multiplied into its square will produce the number proposed.

RULE. 1. Separate the given number into periods of three figures each, beginning at the units place. These periods will denote the number of figures the required root will contain.

2. Find the greatest root contained in the left hand period, which place to the right of the given number; subtract the cube of this root from the said period, and to the remainder bring down the next period for a dividual.

3. Square the root and multiply the square by 3 for a defective divisor.

4. Reserve mentally the units and tens of the dividual, and try how often the defective divisor is contained in the rest ; place the result of this trial to the root, and its

square to the right of said divisor, supplying the place of tens with a cipher, if the square be less than ten.

5. Complete the divisor by adding thereto the product of the last figure of the root by the rest and by 30.

6. Multiply and subtract as in simple division, and bring down the next period for a new dividual ; for which find a divisor as before, and so proceed with every period.

Note. When there are decimals in the given number, separate the periods both ways from the units place, annexing as many ciphers to the decimals as may be deemed necessary. The root must consist of as many whole number and decimal figures as there are periods of each in the given number.

PROOF. Involve the root to the third power, adding the remainder, if any, to the result.

EXAMPLES 1. What is the cube root of 99252.847 ?

99252.847(46.3. 64

Defective divisor and square of 6: 4836)35252 + 720'= complete divisor

5556)33336 Defective divisor & square of 3 = 634809)1916847 + 4140 = complete divisor 638949)19 16847

2. What is the cube root of 16194277 ? Ans. 253. 3. What is the cube root of 389017?

Ans. 73. 4. What is the cube root of 5735339 ? Ans. 179. 5. What is the cube root of 34328125 ? Ans. 325. 6. What is the cube root of 22069810125? Ans. 280.5 7. What is the cube root of 12.977875 ? Ans. 2.35 8. What is the cube root of 36155.027576? Ans. 33.06+ 9. What is the cube root of 15926.972504? Ans. 25,16+ 10. What is the cube root of .001906624? Ans. .124

Note 1.-The cube root of a vulgar fraction is found by reducing it to its lowest terms, and extracting the root of

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