4. What is the square root of 6561 ? Ans. 81. Ans. 667. Ans. 997. Ans. 5464. Ans. 59049. 5. The cipher in the trial divisor may be omitted, and its place, after division, occupied by the trial root figure, thus forming in succession only complete divisors. 10. What is the square root of 2? 11. Extract the square roots of the following numbers: 13. What is the square root of .0043046721 ? Ans. .06561. 6. The square root of a common fraction may be obtained by extracting the square roots of the numerator and denominator separately, provided the terms are perfect squares; otherwise, the fraction may first be reduced to a decimal. 7. Mixed numbers may be reduced to the decimal form before extracting the root; or, if the denominator of the fraction is a perfect square, to an improper fraction. 14. Extract the square root of 6251 Ans.. Ans..816496+. APPLICATIONS. 435. An Angle is the opening between two lines that meet each other; thus, the two lines, A B and A C meeting, form an angle at A. 436. A Triangle is a figure having three sides and three angles, as A, B, C. 437. A Right-Angled Triangle is a triangle having one right angle, as at C. 438. The Base is the side on which it stands, as A, C. 439. The Perpendicular is the side forming a right angle with the base, as B, C. B 440. The Hypotenuse is the side opposite the right angle, as A, B. 441. Those examples given below, which relate to triangles and circles, may be solved by the use of the two following principles, which are demonstrated in geometry. 1st. The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. 2d. The areas of two circles are to each other as the squares of their radii, diameters, or circumferences. 1. The two sides of a right-angled triangle are 3 and 4 feet; what is the length of the hypotenuse? OPERATION. 329, square of one side. 42 = 16, square of the other side. 25, square of hypotenuse. √/25= 5, Ans. ANALYSIS. Squaring the two sides and adding, we find the sum to be 25; and since the sum is equal to the square of the hypotenuse, we extract the square root, and obtain 5 feet, the hypotenuse. Hence, To find the hypotenuse. Add the squares of the two sides, and extract the square root of the sum. To find either of the shorter sides. Subtract the square of the given side from the square of the hypotenuse, and extract the square root of the remainder. EXAMPLES FOR PRACTICE. 2. If an army of 55225 men be drawn up in the form of a square, how many men will there be on a side? Ans. 235. 3. A man has 200 yards of carpeting 14 yards wide; what is the length of one side of the square room which this carpet will cover? Ans. 45 feet. 4. How many rods of fence will be required to inclose 10 acres of land in the form of a square ? Ans. 160 rods. 5. The top of a castle is 45 yards high, and the castle is surrounded by a ditch 60 yards wide; required the length of a rope that will reach from the outside of the ditch to the top of the castle. Ans. 75 yards. 6. Required the height of a May-pole, which being broken 39 feet from the top, the end struck the ground 15 feet from the foot. Ans. 75 feet. 7. A ladder 40 feet long is so placed in a street, that without being moved at the foot, it will reach a window on one side 33 feet, and on the other side 21 feet, from the ground; what is the breadth of the street? Ans. 56.64+ ft. 8. A ladder 52 feet long stands close against the side of a building; how many feet must it be drawn out at the bottom, that the top may be lowered 4 feet? Ans. 20 feet. 9. Two men start from one corner of a park one mile square, and travel at the same rate. A goes by the walk around the park, and B takes the diagonal path to the opposite corner, and turns to meet A at the side. How many rods from the corner will the meeting take place? Ans. 93.7+rods. 10. A room is 20 feet long, 16 feet wide, and 12 feet high; what is the distance from one of the lower corners to the opposite upper corner? Ans. 28.284271+ feet. 11. It requires 63.39 rods of fence to inclose a circular field of 2 acres; what length will be required to inclose 3 acres in circular form? Ans. 77.63+ rods. 12. The radius of a certain circle is 5 feet; what will be the radius of another circle containing twice the area of the first? Ans. 7.07106+ feet. CUBE ROOT. 442. The Cube Root of a number is one of the three equal factors that produce the number. Thus, the cube root of 27 is 3, since 3 × 3 × 3 = 27. 443. In extracting the cube root, the first thing to be determined is the relative number of places in a cube and its root. The law governing this relation is exhibited in the following examples: From these examples, we perceive, 1st. That a root consisting of 1 place may have from 1 to 3 places in the cube. 2d. That in all cases the addition of 1 place to the root adds three places to the cube. If we point off a number into three-figure periods, commencing at the right hand, the number of full periods and the left hand full or partial period will indicate the number of places in the cube root, the highest period answering to the highest figure of the root. 444. 1. What is the length of one side of a cubical block containing 413494 solid inches? OPERATION-COMMENCED 413494 74 343 14700 70494 ANALYSIS. Since the block is a cube, its side will be the cube root of its solid contents, which we will proceed to compute. Pointing off the given number, the two periods show that there will be two figures, tens and units, in the root. The tens of the root must be extracted from the first period, 413 thousands. The greatest cube in 413 thousands is 343 thousands, the cube of 7 tens; we therefore write 7 tens in the root at the right of the given number. Since the entire root is to be the side of a cube, let us form a block, (Fig. 1), by the addition of 70494 solid inches, in such a manner as to preserve the cubical form, its size will be that of the required block. To preserve the cubical form, the addition must be made upon three adjacent sides or faces. The addition will therefore be composed of 3 flat blocks to cover the 3 faces, (Fig. II); 3 oblong blocks to fill the vacancies at the edges, (Fig. III); and 1 small cubical block to fill the vacancy at the corner, (Fig. IV.) Now, the thickness of this enlargement will be the additional length of the side of the cube, and, consequently, the second figure in the root. To find thickness, we may divide solid contents by surface, or area. But the area of is 70 × 70 × 3 = 14700 square inches. This number is obtained in the operation by annexing 2 ciphers to three times the square of 7; the result being written at the left hand of the dividend. Dividing, we obtain |