Note. When more than half the root is found, the remaining gures of it may be found by divifion, making ufe of the laft divifor, and bringing down fo many of the next figures of the refolvend, as there were periods to come down, when you began the divifion. RULES for the SQUARE ROOT of VULGAR FRACTIONS and MIXED NUMBERS. After reducing the fraction to its loweft terms, for this and all other roots; then, I. Extract the root of the numerator for a new numerator, and the root of the denominator for a new denominator, which is the beft method, provided the denominator be a complete power. But if it be not, 2. Multiply the numerator and denominator together; and the root of this product being made the numerator to the denominator of the given fraction, or made the denominator to the numerator of it, will form the frac tional part required :—Or, 3. Reduce the vulgar fraction to a decimal, and ex. tract its root. 4. Mixed 4. Mixed numbers may either be reduced to improper fractions, and extracted by the firft or fecond rule; or the vulgar fraction may be reduced to a deci. mal, then joined to the integer, and the root of the whole extracted. EXAMPLES. 1: What is the fquare root of 445? 144 16 By Rule 1. 1+1= 16(4 root of the numerator.' 1512 16 1681(41 root of the denominator. 16 81)81 Therefore the root of the given fraction. 81 Q.4 By Rule z. 4 16×1681=26896 and ✔ 26896=164. Then, 1681) 16,0095181439+. And ,0095181439= ,09756+. 793 2. What is the fquare root of 12? 3. What is the fquare root of 42 ? Anfw. 77. Note. In extracting the fquare or cube root of any furd number, there is always a remainder or fraction left, when the root is found: To find the value of which, the common method is, to annex pairs of cyphers to the refolvend for the fquare, and ternaries of cyphers to that of the cube, which makes it tedious to difcover the value of the remainder, especially in the cube. Now this trouble may be faved by the following method. In the fquare, the quotient is always doubled for at new divifor Therefore, when the work is completed, the root doubled is the true divifor, or denominator* to its own fraction; as, if the root be 12, the denominator will * These denominators give a small matter too much in the fquare root, and too little in the cube, yet they will be fufficient in common use, will be 24; to be placed under the remainder; which vulgar fraction, or its equivalent decimal, must be annexed to the quotient, or root, to complete it. If to the remainder either of the fquare or cube, cyphers be annexed, and divided by their refpective denominators, the quotient will produce the decimals belonging to the root. APPLICATION and USF of the SQUARE ROOT. PROBLEM I. To find a mean proportional between two numbers. RULE.-Multiply the given numbers together, and extract the fquare root of the product; which root will be the mean proportional fought. EXAMPLES. What is the mean proportional between 24 and 96 ? 96×24 48 Anfw. PROBLEM II. To find the fide of a Square equal in Area to any given Superficies whatever. RULE. Find the Area, and the fquare root is the fide of the fquare fought. EXAMPLES. I. If the area of a circle be 184,125, What is the fide of a fquare equal in area thereto ? ✓ 184,125=13,569+ Anf. 2. If the area of a triangle be 160; What is the fide of a fquare equal in area thereto ? 160=12,649+ Anf.. PROB. PROB. III. A certain General has an army of 5625 men; Pray how many muft-he place in rank and file, to form them into a square? √5625=75 Anf.* PROB. IV. Let 10952 men be fo formed, as that the number in rank may be double the file. 10952 74 in file, and 74X2=148 in rank. 2 PROB. V. If it be required to place 2016 men so as that there may be 56 in rank and 36 in file, and to ftand 4 feet diftance in rank, and as much in file. How much ground do they stand on? To answer this, or any of the kind, ufe the following proportion :-As unity to the distance :: fu is the number in rank lefs by one: to a fourth number ;-next, do the fame by the file, and multiply the two numbers together, found by the above proportion, and the product will be the answer.t As 14:56—1 : 220. And as 1 : 4 :: 36—1: 140. Then, 220 X 14030800 fquare feet, the Anf. PROB. VI. Suppose I would fet out an orchard of 600 trees, fo that the length fhall be to the breadth as 3 to 2, and the diftance of each tree, one from the other, 7 yards; How many trees muft if be in length, and how many in breadth; and how many square yards of ground do they ftand on ? Τα * If you would have the number of men be double, triple, or quadruple, &c, as many in rank as in file; extract the fquare root of,,, &c. of the given number of men, and that will be the number of men in file, which double, triple, quadruple, &c. and the product will be the number in rank. + The above rule will be found ufeful in planting trees, having the diftance of ground between each given. : To refolve any queftion of this nature; fay, as the ratio in length is to the ratio in breadth :: fo is the number of trees to a fourth number; whofe fquare root is the number in breadth ;- -And as the ratio in breadth : is to the ratio in length:: fo is the number of trees to a fourth, whose root is the number in length. PROB. VII. Admit a leaden pipe inch diameter will fill a ciftern in 3 hours; I demand the diameter of another pipe, which will fill the fame ciftern in 1 hour. RULE. As the given time is to the fquare of the given diameter, fo is the required time, to the square of the required diameter. =,75; and ,75 X 75,5625: Then, As 3h.:,5625 :: 1h.: 1,6875 inverfely, and 1,68751,3 inch nearly, Anfwer. PROB. VIII. If a pipe, whofe diameter is 1,5 inch, fill a ciftern in 5 hours; In what time will a pipe, whofe diameter is 3,5 inches fill the fame ? 1,5 × 1,5=2,25; and 3,5 × 3,5=12,25: Then, As 2,25:5:: 12,25,91 hour, inverfely, 54 min. 36 fec. Anfwer. PROB. IX, If a pipe, 6 inches bore, will be 4 hours in running off a certain quantity of water; In what time will 3 pipes, each 4 inches bore, be in difcharging double the quantity? 6×6=36, 4X4≈16, and 16x3=48. Then, as 36* 4h.:: 48: 3h. inversely, and as 1w.: 3h. :: 2w.: 6h. Anfwer. PROB. |