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1. What is the square root of 1225 ° Operation.

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Explanation.—9 is the greatest square in 12 the left hand period, we therefore subtract 9 from the period 12, and place the root, which is 3, to the right hand for the first figure of the root. We then bring down 25 the next period, and double 3 the first figure of the root for the left hand figure of the divisor. 6 can be taken from 32 five times, we therefore place 5 for the next figure of the root, and to the right hand of the divisor, then multiply the whole divisor by the last figure in the root.

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3. What is the square root of 55225 * - Ans. 235. 4. Required the square root of 452929. Ans. 673. 5. Required the square root of 1234321. Ans. 1111.

JNote 1.--When the divisor cannot be taken from all the figures except the right hand one, a cipher must occupy the next place in the root.

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7. What is the square root of 102'030'2012 Ans. 10101.

JWote 2.—If there be a remainder, the operation may be continued by annexing two ciphers to each remainder, and the figures placed in the quotient, after the ciphers are annexed, will be decimals.

8. What is the square root of 2672 Ans. 51,691 --.

Illustration of the principles on which the foregoing method of extracting the square root, is founded.

When any number of square surfaces are placed in the form of a square, the number of small squares on one side, that is, the length of the whole square, is the square root of the whole number of small squares given. (See page 252.) Therefore in extracting the square root of a number, we are to find a number, which multiplied into itself, will produce the given number; or otherwise, we are to find the length of one side of a square, when the contents of that square are given.

Let us suppose we have 169 tiles,” each 1 foot square, and that we wish to know how long the side of a square platform would be, made by the same. To ascertain the length of one side of the supposed platform, we must extract the square root of 169, the whole number of tiles, for 169 would be the contents of the whole square, and the length of one side, the root.

* A tile is a large brick having a square form.

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Since the whole number consists of 3 places of figures, there must be two places in the root. The left hand period is 1, or as it stands in the third place, is 100, and the greatest square in 100 is 100, and the square root of 100 is 10. We subtract the square, 100 and place 10 the root to the right hand, as may be seen in the first Fig. 1. part of the operation. We have now taken away from the whole number of supposed tiles 100, which may be supposed to be arranged into a square form as in fig. 1. This fig is composed of 100 small squares, and has 10 the square root of 100, on each side. If we multiply 10 , I " - by itself, the product would be 100, that part of 169 of which we have found the root. The 69 tiles which remain, must be placed on two sides of the square already formed, which meet in the same point, so that the form of the whole may still be a square. In the full operation, we find the root of the left hand period first, and call it 1, but which is actually 10, since another figure is to be placed to the right of it. By subtracting the square of 1 from the left hand period, we actually take 100 from the whole number, since we take it from the third place. Now if we add one row to two sides of fig. 1st, the number of tiles required will be twice as many as that part of the root already found, and as the first figure of the root is 10 the number of tiles required to make one row on a side of figure 1st, will be 10, and to make a row on two sides the number required will be 20, twice 10. Therefore, as many times as we can take 20, double the first figure of the root, from the number of tiles remaining, so many rows each consisting of 10 tiles, can we add to each of two sides. This explains the reason for doubling the root for a divisor. In fig. 2d, let the part A, be just as large as the whole of figure 1st, then the parts - a b and a c will

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Fig. ii. represent the number of tiles it will require to add three rows to each of two sides of a figure as large as fig. 1. which is 60, just three times 20, the divisor found by doubling the root in the first part of the operation. As 20 Jo the first part of the divisor can be taken 3 times from 69, the remaining number of tiles, the next figure in the root must be 3. Multiplying 20 by 3, the product expresses tiles enough to complete the parts a b, a c, but this does not leave the figure in the form of a square, the corner a is still deficient. The side of the small square a, which is still wanting in figure 2d, will always be the same number as the number of rows added to two sides of A, and Fig. 111. if we square the - number of rows § added to one side, the square will & f express as many tiles as the small square a will contain. A - - In the full ope- ration, before we multiply 20 by 3, the last figure in the root, we place , the 3 to the right * of the 2 tens, and multiplying the 3 by the 3 in the root, gives tiles enough to fill the corner a, and multiplying the 2 tens by the 3, gives enough to fill the parts a b and a c, as may be seen in fig. 3, where the part A is supposed to be equal to the whole of fig. 1. In fig. 3, the length of c f is represented by 10, the first figure of the root, and fg by 3, the second figure, consequently, c g must be 13, which shows that the number of tiles on one side of the platform, would be 13, that is, the length of one side, 13 feet. - o Now if we multiply 13 feet by 13 feet, the product will be 169 feet, the number of tiles given; which shows that the root has been correctly extracted. If there were another period of figures to bring down, we should proceed in the same manner as we have in the last period, and add to two sides of fig. 3, as we now have to fig. 1. As the length of the side of a square multiplied into itself, produces the contents of the square, or the second power of the root, the learner can easily determine whether he has worked the question correctly, by squaring the root, which will give the given number, if the operation has been rightly performed. If there be a remainder, the root is not exact ; to obtain the power in that case, the remainder must be

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