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The root of ### is nearest to #4 = 1+. This differs from the true root by a quantity less than to. If greater exactness is required, a number larger than 144 may be used.
. What is the approximate root of # 3 . What is the approximate root of ### 7. What is the approximate root of 33 : 8. What is the approximate root of 17?, ? 9. What is the approximate root of 3? 10. What is the approximate root of 7 ? 11. What is the approximate root of 417 ?
The most convenient numbers to multiply by, in order to approximate the root more nearly, are the second powers of 10, 100, 1000, &c., which are 100, 10000, 1000000, &c. By this means, the results will be in decimals.
To find the root of 2 for instance, first reduce it to hundredths.
2 = ###, the approximate root of which is ## = 1.4.
Again 2 = #####, the approximate root of which is ### = 1.41.
Again, 2 = }}#####, the approximate root of which is #### = 1.414.
In this way we may approximate the root with sufficient accuracy for every purpose. But we may observe, that at every approximation, two more zeros are annexed to the number. In fact, if one zero is annexed to the root, there must be two annexed to its power; for the second power of 10 is 100, that of 100 is 10000, &c.
This enables us to approximate the root by decimals, and we may annex the zeros as we proceed in the work, always annexing two zeros for each new figure to be found in the root, in
the same manner as two figures are brought down in whole numbers.
The root of 2 then may be found as follows.
12. What is the approximate root of 28 °
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
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.
16. What is the approximate root of 213.53?
XXX. Questions producing pure Equations of the Second - #.
1. A merger 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 f
Let w = the numoer of shillings he paid per yard.
Then o: = the number of yards.
The price of the whole will be ** = 324 shillings,
.ins. 27 yards, at 12s. per yard.
2. A detachment of an army was marching in regular cohumn, 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.
Leta = the number in front; then r + 5 = the number in depth; a" + 5 v = the whole number of men. Again r + 845 = the number in front after the movement; And 5 a + 4225 = the whole number.
The number of men = 5 a + 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 2
.Ans. The side is 12.649 -t- rods. It cannot be found exactly, because 160 is not an exact 2d power.
This is exact within less than roorg of a rod. It might be carried to a greater degree 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 P
- - - z 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 14 dollars. What was the sum to be divided ?
Let a = the number of dollars to be divided:
then ** = the number of men resident, and also the sum each received. * The root of a is properly expressed by the fractional 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 performed, as respects that letter, by
adding the exponents. Thus a × a = a + = a”; a × wo = a + ' = a”, &c. Applying the same rule; if .# represents a root or first power, the second power or * X o F. ** + # "= a-' or w. * 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 a” x a' = a +* = a ** = a”. This furnishes us with a simple rule to find the root of a literal quantity; which is, to divide its exponent by 2.
a = z* = the number of dollars each received. and I. T.; o a; .* + 2 = the number of men the succeeding week; —f--- the number of dollars each received the latter week; * +2 Hence by the conditions