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Hence it appears, that when an example involves the second power of the unknown quantity, the value of the second power must first be found in the same manner as the unknown quantity is found in simple equations; and from the value of the second power, the value of the first power is derived.
It is easy to find the second power of any quantity, when the first power is known, because it is done by multiplication; but it is not so easy to find the first power from the second. It cannot be done by division, because there is no divisor given. When the number is the second power of a small number, the first power is easily found by trial, as in the above examples. When the number is large, it is still found by trial; but a rule may be very easily found, by which the number of trials will be reduced to very few. The first power is called the root of the second power, and when it is required to find the first power from the second, the process is called extracting the root.
It has been shown, Art. XXIV. that the second power of every quantity, whether positive or negative, is necessarily positive; thus 3 X 35+9, and also
3 +9. So a Xa=a?, and also а Х a’. Hence every second power, properly speaking, has two roots, the one positive and the other negative. The conditions of the question will generally show which is the true answer.
Extraction of the Second Root.
In order to find a rule for extracting the root, or finding the first power from the second, it will be necessary, first, to observe how the second power is formed from the first. Let a =
20 and 6 7; then a +b= 27. The second power of a + b is (a + b) (a + b)= a + 2ab + b*
20 X 20 = 400 a b
= 20 X 7= 140 ab= 20 X 7 140
62 7 X 7= 49 a + 2 ab + b = 729.
The product is formed in precisely the same manner in the usual mode of multiplication, as may be seen, if the products are written down as they are formed, without carrying.
49 140 140 400
729 Here we observe, 7 times 7 is 49, 7 times 20 is 140, 20 times 7 is 140, and lastly 20 times 20 is 400. These added together make 729, which is the second power of 27.
1st. When the root or first power consists of two figures, the second power consists of the second power of the tens, plus the product of twice the tens by the units, plus the second power of the units.
2d. The second power of 9, the largest number consisting of one figure, is 81; and the second power of 10, the smallest mumber consisting of two places, is 100; and the second power of 100, the smallest number consisting of three places, is 10000. Hence, when the root consists of one figure, the second power cannot exceed two figures; and when the root consists of two figures, the second power consists of not less than three figures, nor more than four figures.
From these remarks it appears, that we must first endeavor to find the second power of the tens, and that it will be found among the hundreds and thousands.
Let it be required to find the root of 729. This number contains hundreds, therefore the root will contain tens. cond power of the tens is contained in the 700. 20 x 20 is 400, and 30 X 30 is 900. 400 is the greatest second power of tens contained in 700. The root of 400 is 20. Subtract 400 from 729, and the remainder is 329. This must contain 2ab + b?, that is, the product of twice the tens by the units, plus the second power of the units. If it contained exactly the
product 2 a b of twice the tens by the units, the units of the root
729 (20 +7= 27 root.
329 (40 divisor.
18,49 (40 +3=43 root.
249 (80 divisor. 43 X 43= 1849. In this example, the second power of the tens will be found in the 1800. 30 x 30 900; 40 X 40 = 1600; 50 X 50 = 2500. The greatest second power in 1800 is 1600, the root of which is 40. Write 40 in the place of a quotient. Subtract 1600 from 1829. The remainder is 249, which divided by twice 40, or 80, gives 3. Add 3 to the root, and raise the whole to the second power. 43 X 43 = 1849. Therefore 43 is the root required.
It is evident that the result will not be affected, if instead of writing 40 in the root at first, we omit the zero, and then subtract the second power of 4, viz. 16 from the 18, omitting the two zeros which come under the other period. Then to form the divisor, the 4 may be doubled, and the divisor will be 8, instead of 80, and the dividend must be 24, the right hand figure being rejected.
Examples. 1. What is the root of 1444?
Ans. 38. 2. What is the root of 7396? 3. What is the root of 361 ? 4. What is the root of 3249? 5. What is the root of 7921 ? 6. What is the root of 8281?
The second power of a +b+c, or (a+b+c) (a+b+c) is a2 + 2ab + b2 + 2 ac + 2bct ca
a + 2 ab + b3 + 2 (a + b) c + c?.
4356 1452 5082
The first three terms of the formula, viz:
a + 2ab + b?, are the second power of a + b or of the hundreds and tens, viz. 720. The second power of 720 can have no significant figure below hundreds, and the significant figures of the second power of 720 and of 72 are the same; the former is 518400, the latter 5184. If from the whole number 527076 the two right hand figures be rejected, the number is 5270. This contains the second power of 72 and something more, viz. a part of the product 2 X (700 + 20) X 6= 2 (a + b) c.
The method of procedure then, is to find the largest root contained in 5270. The first three terms of the above formula, viz. a + 2ab + b?, show, that this is to be found by the method given above for finding a root consisting of two figures.
37,0 (14 72 X 72 = 51,84
The root is 72, and the remainder is 86. Annex to this the two figures rejected above, and it becomes 8676. This contains 2 (a + b)c + c'; that is,
2 X 720 X c + c*. If 8676 be divided by 2 X 720 = 1440, the quotient will be either c or a number larger by 1 or 2. The zero on the right of 1440, and the right hand figure in the dividend may be omitted without affecting the quotient. The quotient is 6. Put 6 into the root and raise the whole to the second power.
726 X 726 = 527076