Nose. When more than half the root is found, the remaining #gures of it may be found by division, making use of the last divisor, and bringing down so many of the next figures of the refolvend, as there were periods to come down, when you began the division. Rules for the SQUARE Root of VULGAR FRACTIONS and Mixed NUMBERS. I. After reducing the fraction to its lowest terms, for this and all other roots ; then, Extract the root of the numerator for a new nume. rator, and the root of the denominator for a new denomi. nator, which is the best method, prosi', the denominator be a complete power. But if ii 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 fractional part required :-Or, 3. Reduce the vulgar fraction to a decimal, and ex. tract its rooca 4. Mixed 4. Mixed numbers may either be reduced to improper fractions, and extracted by the first or second 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. By Rule 1. 16 1. 16 1681(41 root of the denominator. 16.4 16 2793 81)81 Therefore 4=the root of the given fraction, By Rule 2. == ==,09756+ By Rule 3. 1681)16(,009518 1439+. And V,0095181439 = ,09756+ 2. What is the square root of zz?} ? Answ. i'l. 3. What is the square root of 42 ? Answ. 6s. Note. In extrading the square or cube root of any surd 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 resolvend for the square, and ternaries of cyphers to that of the cube, which makes it tedious to discover the value of the remainder, especially in the cube. Now this trouble may be saved by the following method. In the square, the quotient is always doubled for a new divisor : Therefore, when the work is completed, the root doubled is the true divisor, or denominator* to its own fraction; as, if the root be 12,. the denominator will * These denomipators give a small matter too much in the {quare root, and too little in the cube, yet they will be fufficiene in common use. T 2 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 square or cube, cyphers be annexed, and divided by their respective denominators, the quotient will produce the decimals belonging to the root. APPLICATION and Use of the SQUARE ROOT. PROBLEM. I. To find a mean proportional between two numbers. RULE.--Multiply the given numbers together, and extract the square root of the product ; which root will he the mean proportional fought. E x A M P L E S. What is the mean proportional between 24 and 96 ? ✓ 96 X 24=48 Answ. PROBLEM 11. To find the side of a Square equal in Area to any given Superficies whatever. Rule.- Find the Area, and the square root is the fide of the square fought. Ε Χ Α Μ Ρ Ι Ε s. a If the area of a circle be 184,125, What is the fidė of a square equal in area thereto? ✓ 184,125=13,569+ Ang 2. If the area of a triangle be 100; What is the side of a square equal in area thereto? 160=12,649+ Anf. a PROB. Prob. III. A certain General has an army of 5625 men ; Pray how many must he place in rank and file, to form them into a square ? V 5625=75 Anfi* = Prob. IV. Let 10952 men be fo formed, as that the number in rank may be double the file. 10952 =74 in file, and 74 X 2=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 stand 4 feet distance in rank, and as much in file. How much ground do they stand on ? To anfwer this, or any of the kind, use the following proportion :-As unity : to the distance : : fu is the number in rank less by one : to a fourth number ;--next, do the same by the file, and multiply the two numbers together, found by the above proportion, and the product will be the answer.t As 1 : 4:: 56–1:220. And as 1 : 4 :: 36-1: 140. Then, 220 X 140=30800 square feet, the Anf. PROB. VI. Suppose I would set out an orchard of 600 trees, so that the length fhall be to the breadth as 3 to 2, and the distance of each tree, one from the other, 7 yards ; How many trees must it be in length, and how many in breadth; and how many square yards of ground do they ftand on? Tö * If you would have the number of men be double, triple, or quadruple, &c, as many in rank as in file ; extract the square 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 distance of ground between each given, to 133. To resolve any question of this nature ; fay, as the ratio in length : is to the ratio in breadth :: so is the number of trees : to a fourth number ; whose square 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. As 3 : 2 :: 600 : 400. And V 400=20= number in breadth. As 2 : 3 :: 600 : 900. And v 90o=30= number in length. As 1:7:: 30-1 : 203. And as 1:7:: 20-1; And 203 X 133 =26999 square yards, the Anfwer. PROB. VII. Admit a leaden pipe 3 inch diameter will 2 fill a cistern in 3 hours ; I demand the diameter of another pipe, which will fill the same ciftern in 1 hour. Rule...As the given time is to the square of the given diameter, fo is the required time, to the square of the required diameter. *=,75 ; and 275 X 755,5625 i Then, As 3h. : ,5625 :: 1b. : 1,6875 inversely, and 1,6875=1,3 inch nearly, Antwer. Prob. VIII. If a pipe, whose diameter is 1,5 inch, fill a cistern in 5 hours ; In wkat time will a pipe, whose diameter is 3,5 inches fill the same ? 1,5 x 1,5=2,25 ; and 3,5 x 3,5=12,25 : Then, As 2,25 : 5 :: 12,253,91 hour, inversely,=54 min. 36 sec. Answer. 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 discharging double the quantity ? 6x6=36, 4X4=16, and 16 x 3=48. Then, as 36 4h. :: 48 : 3h. inversely, and as iu. : 3h. :: 2w.: 6h. Answer. PROB. a : |