EXAMPLES. Let 2 be raised to all the successive powers, from the 2 × 2 × 2 × 2 = 16 biquadrate or 4th power. 2 × 2 × 2 × 2 × 2 32, = 5th power. 1. A floor is 12 feet square, how many feet of boards does it contain? Ans. 144 ft. Ans. 90000. 2. In a plantation 300 perches square, how many perches does it contain? TABLE OF POWERS. Roots Squares. Cubes. 4th pow. 5th pow. 6th pow. 7th pow. 8th pow. 9th pr. 1. To extract the Square Root prepare the number for extraction by pointing it from units place, into periods of two figures each. 2. Find by involution or for convenience sake in the table, a square nearest to the first, subtract and bring down the next period, which place to the right of the remainder for a dividend. 3. Double the quotient figure for a divisor, and try how often it is contained in all the figures of the dividend except the one, on its right. 4. Place this in the quotient for a second figure of the root, as well as to the right of the divisor. 5. Multiply by this quotient figure as in division, the product subtract as before, and to the difference bring down the third period. 6. Proceed in like manner, still doubling the quotient figures for a new divisor, and bringing down another period each time for a new dividend, until the whole is completed. EXAMPLE. 1. What is the square root of 10, 34, 26, 56? (3216 9 62)134 124 641)1026 641 6426)38556 2. Extract the square root of 151321? Ans. 389 rt. In extracting the root of whole numbers and decimals, one half the number of decimal figures must be pointed out in the quotient. 1. What is the square root of 3271.4007? Ans. 57.19 + 2. 66 66 of 4.372594? 66 2.091 + 66 66 CASE III. To extract the Square Root of a mixed number. RULE 1.-Reduce the fractional part of the mixed number to its lowest term, and then the mixed number to an improper fraction. 2. Extract the roots of the numerator and denominator for a new numerator and denominator. 3. If the mixed number given be a surd, reduce the fractional part to a decimal, annex it to the whole number, and extract the square root. 1. What is the square root of 421? 2. What is the square root of 51%}? Ans. 61. 7}. CASE IV. To extract the Square Root of a fraction. RULE. Reduce the fraction to its lowest terms, then extract the square root of the numerator for a new numerator, and the denominator for a new denominator. 1. What is the square root of 2. What is the square root of 2 CASE V. To extract the Square Root of a Surd. Ans. 3. 66 4. RULE. Reduce the surd to a decimal, and extract the root thereof. 1. What is the root of 2. What is the root of 2? CASE VI. Ans. 89802 + 66 86602 + To find a mean proportional. RULE.-Multiply the two numbers together, and extract the square root of the product for a mean proportional. 1. Suppose A. in a school room sits 4 feet from a hot stove, and B. 9 feet from the same, how much warmer is A. than B? Ans. 6 times. 2. Two ships sail from the same port, one goes due North 128 miles, the other due East 72, how far are the ships asunder? Ans. 146.86 miles. CASE VII. The base and perpendicular given to find the hypothenuse. 1. The top of a castle is 45 yards high, and is surrounded with 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 yds. 2. The wall of a fort is 25 feet high, which is surrounded by a moat 30 feet in breadth, I want to know the length of a ladder that will reach from the outside of the moat to the top of the wall? Ans. 39.05 ft. CASE VIII. To find the distance that any object may be seen at sea, elevated at any height above the level of the water. RULE. Add to the earth's semi-diameter in feet, the height of the object; square the sum, next square the number of miles in the earth's semi-diameter, take the difference of those squares, then by (Euclid 47, Lib. 1,) the square root of the difference of those squares will be the distance required in feet. 1. There is a point of the Andes in South America which is 4 miles above the level of the sea, to what distance could a person see from the top of such an elevated point, provided the atmosphere was perfectly clear, and not assisted by refraction. Illustration.-If we put the earth's semi-diameter at 4000 miles, then 4004 x 4004 4000 x 4000 = 178.93 miles, the distance required, which is about twice as far as a person could see elevated from a point one mile above the level of the sea. 2. 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. Cubes. 1 8 27 64 125 216 343 512 729 RULE 1.-Point every third figure beginning at the units place, then find the nearest cube to the first point, and subtract it therefrom; put the root in the quotient, bring down the figure in the next point to the remainder for a dividend. 2. Square the quotient and multiply it by 3, for a divisor; find how often the divisor is contained in the dividend, rejecting units and tens, and place the number of times in the quotient. 3. Square the last figure placed in the quotient, and place the result to the right hand of the divisor, for a defective divisor. 4. Then multiply the last figure placed in the quotient by the other figures, and that product by 30; add the last product to the (defective divisor,) placing units under units and tens under tens, for a complete divisor. Illustration.-Extract the cube root of |