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NOTE. Double the square root is the denominator of the remaining fraction, thus, 339.

41381

6. Required the square root of 1296: Ahs. 36. 7. What is the square root of 360000 ? 8. What is the square root of 137641 ? 9. Required the square root of 123692,89.

10. Required the square root of 138387.

Ans. 600.
Ans. 371..

Ans. 351,7.

Ans. 372+

11. What is the square root of 408326? Ans. 639+..

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The square root is useful in geometry, measuring of superficies, surveying, &c.

12. A surveyor is required to lay out 1000 acres of land, in form of a square, how many rods will be the length of each side? Ans. 400 rods..

NOTE. The 12 example is solved by reducing the acres to rods, and then extracting the square root of those rods..

13. There is a ditch 20 feet wide, full of water, on one side of the ditch is a wall 15 feet high from the

water; it is required to make a ladder that will just reach the top of the wall from the opposite side of the ditch; how long must it be? Ans. 25 feet.

To find the length of the ladder, add the square of the width of the ditch, and the square of the heighth of the wall together, and extract the square root of their sum.

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To extract the square root of a vulgar fraction.

RULE. Reduce the fraction to its lowest term, then extract the square roots of the numerator, and also of denominator, and set them down fractionwise, it will be the root required.

NOTE. The same rule holds good for extracting the cube or any other root of a vulgar fraction.

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The cube root of a number is that number, which if multiplied by itself, and that product multiplied by the first number, the second product will be the cube, or number of which the cube root is required, stre

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The given number must be prepared for extraction by placing a point over the unit figure, and every third figure each way from it, so as to have periods of three figures each.

There will be the same number of figures in the root, as there are periods, either of whole numbers, or of decimals.

When there are decimals given, and do not make complete periods of three figures each, annex ciphers sufficient to complete the right hand period.

The first figure sought in the root, is the root of the greatest cube contained in the first or left hand period, and (for distinction) is called a, which is placed in the quotient. The cube of this figure is to be subtracted from the first period.

To the remainder bring down the next period for a new resolvend. Then to 300 times the square of a, ádd 30 times a, their sum will be a divisor, by which the resolvend is to be divided to find the next figure in the quotient, or root, which next figure is called e. Then to 300 times the square of axe, add 30 times the square of exa, and also the cube of e; their sum will be a subtrahend, which place under the resolvend, and subtract it therefrom.* To the remainder bring down the next period, which will make another resolvend. All the figures now in the quotient or root (taken together) are called a, with which proceed as before, to find a divisor. With this new divisor find another quotient (or root) figure, which calle. Proceed with the new a and e as before, to find a subtrahend, which place under the last resolvend, and subtract it therefrom. To the remainder bring down the next period, and proceed as before, calling all the figures now in the root a, to find a divisor, and thus proceed till the work is done.

streapres EXAMPLES.

1. Required the cube root of 46656.

* Sometimes the figure found by dividing as above will make the subtrahend greater the resolvend, in which case a less number must be taken as in the square root.

10083 70 2001 advo qui bonapail a

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2. Required the cube root of 153990656. Ans. 536. 3. What is the cube root of 47652901988,747336.

Ans. 3625,46.

4. Required the cube root of 68921.

Ans. 4

Ans. 6,4.

Ans. 41,1.
Ans. ,09.

5. Required the cube root of 262,144. 6. Required the cube root of 69426,531. 7. Required the cube root of 3000729. 8. What is the cube root of ,000000343? Ans. ,007.

A general rule for extracting the roots of all powers.

1. Prepare the given number for extraction, by pointing off from the unit place, as the root required directs.

2. Find the first figure in the root by your own judgment, or by inspection into the table of powers.

3. Subtract it from the given number.

4. Augment the remainder by the next figure in the given number, that is, by the first figure in the next point, and call this your dividend.

5. Involve the whole root last found, into the next inferior power to that which is given.

6. Multiply it by the index of the given power, and call this your divisor.

7. Find a quotient figure by common division, and annex it to the root.*

8. Involve all the root, thus found, into the given power.

9. Subtract this power (always) from as many points of the given power as you have brought down, beginning at the lowest place.

10. To the remainder bring down the first figure of the next point for a new dividend.

11. Find a new divisor as before, and in like manner proceed till the work is ended.

EXAMPLES.

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What is the cube root of 115501303?

This must sometimes be taken less than the divisor would give, as is often the case in the square and cube roots, by the other method, so as to have the involved root, not exceed the resolvend.

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