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added to the square of that part of the root ob
tained. When the root cannot be exactly expressed by
humbers, it is called a surd.
9. If the contents of a square field be 3844 square rods, what is the length of one side 2 Ans. 62 rods. 10. If a field, in the form of a triangle, contain 8 A. 2 R. 9 r. what would be the side of a square field containing the same number of acres 2
First reduce the whole to square rods. Ans. 37 rods. 11. What is the length of the side of a square field, which contains 10 acres 7 Ans. 40 rods.
v this character set before a number with an index over it, denotes that such a root of the number is to be extracted as is expressed by the index. * implies that the square, or 2d root, of the number after it, is to be extracted; or *v36 stands for the square root of 36. *V denotes the cube root of a number ; and “y the biquadrate or 4th root. The index 2 is generally omitted;—v16=4.
12.% v81 is what number 2 Ans. 9.
Explanation.—It was shown in the Explanation of fig. B, page 244, that when we square # of the length of the side of a square, the square, or 2d power of the root 4, would be # of the whole square, conseQuently, the square root of ; must be #.
It may farther be observed, that in squaring a fraction, we multiply the numerator by itself, and the denominator by itself; and to find the root of a fraction, we must extract the root of the numerator and of the denominator. The square root of 1 is 1, and the square root of 4, 2; so that on both considerations, it appears that the square root of 3 must be 3.
19. What is the square root of Tor 2 Ans. #. 20. What part of the side of a square is the square root of or of the surface of that square 2
.Note.—It is not often that we can extract the square root of a vulgar fraction without a remainder to one or both of the terms; for this reason, the better way is to change the fraction to a decimal before extracting the root.
21. What is the square root of #2 Here #–,42857, and v',42857=,654+the amSWer. 22. What is the square root of ###2 Ans. .412-i-. 23. What fraction is equal to v/#torm 2 Ans. ,0223-1-. 24. What fraction equals v.012 Ans. , 1. 25. If I pay $289 to several men, paying to each man a number of dollars equal to the whole number of men, how many men will there be 2 Ans. 17. 26. If an army, consisting of 24649 men, be drawn up into the form of a square, how many men will there be on each side 2 Ans. 157, 27. A merchant sold cloth to the amount of $37,82}. The number of yards sold equalled the number of cents he received for each yard ; what did he receive per yard 2 Ans. 61 ; cents. 28. The floor of a certain square room can just be covered by 48 yards of carpeting, of a yard wide; what is the length of one side of that room 7 Ans. 18 feet. 29. What number, must be multiplied by itself, that the product may be 82°459°221°458°436 % Ans. 970807706,
30. Of what number is 36°060'025 the square 2
Ans. 6005. 31. The product of what number multiplied by it. self, equals 730,512784 Ans. 27,028. G. In the right angled triangle A B C, the
square of A B c. is equal to the SS square of A C T added to the square of B C ; .A. IB that is, the square A BE D, however large, will be made up of as many square inches .D. L. Il or square feet, &c. as are contained in the squares A I H C and B C G F.”
32. If A C be 9 feet long, and B C 12, what is the length of A B 2
Here 9°-1-12* =225; and v225= 15 feet, the length of A B.
33. If the top of a ladder 20 feet long, rest against the side of a house 16 feet from the bottom of the sills, and the foot of the ladder be on a horizontal line with the sills of the house, how far will the bottom be from the sill 2
The length of the ladder is the hypothenuse, the side of the house the perpendicular end; the ground intervening between the bottom of the ladder and the sill is the base of a right angled triangle, therefore the square of 16, the perpendicular, subtracted from the
* The demonstration of this fact depends upon geometrical principles. The longest side of a right angle triangle, as A B, is named the hypothenuse. One of the shortest sides is called the base, and the other, the perpendicular.
square of 20, the length of the hypothenuse, will leave the square of the distance between the sill and the bottorn of the ladder. Ans. 12 feet.
34. If the ridge of a building be 8 feet from the beams, and the building be 32 feet wide, of what kength must the rafters be 2 Ans. 17,88–H feet. 35. If one end of a ten feet pole be placed 6 feet from the end of a sill, how far from the end of the adjoining sill, must it reach, so that the angle formed by the two sills, shall be a right angle, or to form a square corner 2 - Ans. 8. 36. How long a brace will be required to support a beam, if it be inserted in the post and beam just 4. feet from the point where they meet 2 Ans, 5,65+.
It has been shown that the last product of a number multiplied into itself twice, is the third power or cube of that number; and that the number multiplied into itself twice, is the root or first power. The cube of no single figure can be more than three places of figures, for the cube of 9 the greatest single figure, is only 729. The cube of 99, the greatest number denoted by two figures, is 970'299, six places of figures. The cube of 10, the least number-expressed by two places, is 1000, four places of res. 10 is multiplied into itself twice to produce 1000; 9, twice to produce 729; and 99, twice to produce 970'299; therefore 10 is the cube root of 1000; 9 the cube root of 729, and 99 the cube root of 970°299. If any number of 9's, (which would be the greatest number that could be expressed by the same number of figures,) be cubed, there would not be more than three places of figures for each figure in the root, *
From what has been said, it appears that the cube root of any three figures must always be less than two plases of whole numbers ; also that the cube root of any number of figures more than three and less than seven, must be two places of whole numbers besides the fraction, if any occur. From these facts, it is manifest there must be as many places of whole numbers in the cube root of any number, as there can be periods of three figures each in the given number.
For finding, or extracting the cube root of any number.
1. Point the given number into periods of three figures each, by putting a point over units’ place and every third place towards the left. If there be decimals, point towards the right in the same manner. 2. Subtract the greatest cube contained in the
left hand period from that period, and set the root of that cube to the right of the given number:
Annex the next period to the remainder for a divi
3. Take three times the square of the root already found for an imperfect divisor.” Ascertain how many times the imperfect divisor can be taken from the dividend, and place the number in the root.
4. Multiply three times the whole root, omitting the right hand figure, by the whole root, and add the square of the right hand figure, the sum will form a
Multiply the perfect divisor by the last figure written in the root, and subtract the product from the dividend.
* It is called imperfect, because it is not so large as the real divisor, which cannot be found until the next figure of the root is