What is the approximate root of 28 ? What is the approximate root of 243 ? What is the approximate root of 27068 ? What is the approximate root of 2433 ? The approximate root of which is 15600 15.6, &c. 1000 = But it is plain that this may be performed in the same manner as the above. For if the number 243375000 be prepared in the usual way, it stands thus ; 2,43,37,50,00. Now 243375000 = 243.375000. 1000000 If we take this number and begin at the units and point towards the left, and then towards the right in the same manner, the number will be separated into the same parts, viz. 2,43.37,50,00. The root of this number may be extracted in the usual way, and continued to any number of decimal places by annexing zeros. N. B. The decimal point must be placed in the root, before the first two decimals are used. Or the root must contain one half as many decimal places as the power, counting the zeros which are annexed. What is the approximate root of 213.53? What is the approximate root of 7263 ? What is the approximate root of 17? What is the approximate root of 7? What is the approximate root of 1733 ? XXX. Questions producing pure Equations of the 1. A mercer bought a piece of silk for £16. 4s.; and the number of shillings which he paid per yard, was to the number of yards, as 4 to 9. How many yards did he buy, and what was the price of a yard? Let x = the number of shillings he paid per yard. Ans. 27 yards, at 12s. per yard. 2. A detachment of an army was marching in regular column, with 5 men more in depth than in front; but upon the enemy coming in sight, the front was increased by 845 men; and by this movement the detachment was drawn up in 5 lines. Required the number of men. Let the number in front; then x + 5 = the number in depth; x2 + 5 x = the whole number of men. Again+845 = the number in front after the movement; And 5x+4225 the whole number. x2 + 5 x = 5 x + 4225 x2 = 4225 Ꮖ = 65 The number of men 5x + 4225 = 4550. 3. A piece of land containing 160 square rods, is called an acre of land. If it were square, what would be the length of one of its sides? Ans. The side is 12.649 + rods. It cannot be found exactly, because 160 is not an exact 2d power. This is exact within less than Too of a rod. It might be carried to a greater degere of exactness if necessary. 4. What is the side of a square field containing 17 acres? 5. There is a field 144 rods long and 81 rods wide; what would be the side of a square field, whose content is the same? 6. A man wishes to make a cistern that shall contain 100 gallons, or 23100 cubic inches, the bottom of which shall be square, and the height 3 feet. What must be the length of one side of the bottom? 7. A certain sum of money was divided every week among the resident members of a corporation. It happened one week that the number resident was the root of the number of dollars to be divided. Two men however coming into residence the week after, diminished the dividend of each of the former individuals 1 dollars. What was the sum to be divided? Let the number of dollars to be divided; = then the number of men resident, and also the sum each received. Ꮖ The root of x is properly expressed by the fractiona] index. For it has been observed, that when the same letter is found in two quantities which are to be multiplied together, the multiplication is preformed, as respects that letter, by adding the exponents. Thus a × a = a1+1 = a2 ; x2 × x3 = x2+3 = x3 &c. Applying the same rule; if a represents a root or first power, the second power or x1⁄2 × x × x = x1 + 1⁄2 = x1 or x. x5 The second power of a letter is formed from the first by multiplying its exponent by 2, because that is the same as adding the exponent to itself. Thus as × a3 = a3+3 = a2x3 = a. This furnishes us with a simple rule to find the root of a literal quantity; which is, to divide its exponent by 2. Thus the root of a is aa1; the root of a♦=a3⁄4=a*; 6 a3 the root of a is a = a3 &c. By the same rule, the root of a1 is a; the root of a3 is a; the root of a is a; the root of a" is a", &c. In the above example x = the number of dollars to be divided; |