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denotes that the cube root of 27 is to be extracted; and 1 81=4th root of 81.
When the power is expressed by several numbers, with the signs + or between them, a line or vinculum is drawn from the top of the sign over all the parts of it. Thus, the square root of 30—5 is expressed V 30–5.
A TABLE OF THE SQUARES AND CUBES OF THE NINE DIGITS:
| 1 | 2 | 3 | 41 5 6 7 8 9 1| 4| 9 | 16 | 25 36 49 64
81 118 27 | 64 | 125 1 215 / 343 / 512 | 729
EXTRACTION OF THE SQUARE ROOT. To extract the Square Root of any number, is, to find a number which, being multiplied into itself, shall produce the given number.
1. Point off the given number into periods of two figures each, by putting a dot over the units, another over the place of hundreds, and so on; and if there be decimals, point them in the same manner from units towards the right hand, which dots will show the numbep of figures the root will consist of.
2. Find the greatest square number in the first, or left hand period ; place its root as a quotient in division, and place the square number under the period and subtract it therefrom ; and to the remainder, bring down the next period for a dividend.
3. Double the root already found, and place it at the left hand of the dividend for a divisor.
4. Place such a figure at the right hand of the divisor, and also the same figure in the root, as, when multiplied into the divisor thus increased, the product shall be equal to, or next less than the dividend ; this will be the second figure in the root.
5. Multiply the whole increased divisor by the last figure of the root; place the product under the dividend ,
subtract it therefrom, and to the remainder bring down the next period for a new dividend.
6. Double the figures already found in the root for a new divisor ; and from these find the next figure in the root as last directed; and continue the operation in the same manner until all the periods are brought down.
Note. When there is a deficiency in any period of decimals, you may annex a cipher; or, when there is a remainder, you may continue the operation to decimals, by annexing periods of ciphers.
EXAMPLES. 1. What is the length of one side of a square field which contains 1225 square rods? or what is the square root of 1225 ?
Illustration. By the Rule, we point the given
number into periods of two figures each, by putting 9
a dot over the unit's place and another over the place
of hundreds, making a periods, which show that the 65)325 root will consist of 2 places of figures, viz.: a ten
and a unit. 325
This Rule for determining the number of figures of which the root will consist is founded on 00
the known principle that the places of figures in the
product of any iwo factors cannot exceed the number of places contained in both of those factors, nor can they be but one less than the places in both factors : and as the square root of any power is a factor which, being multiplied into itself, exactly produces that power, consequently, any square number contains just twice as many places of figures as its root, or at least, but one less than twice that number. Then the first period being 12 (hundreds, we seek for the greatest square number that is contained therein, which we find to be 9 (hundreds,) the root of which is 3 (tens=30.) We therefore place 3 (tens) in the root, and its square 9 (hundreds,) under the period 12 (hundreds,) which being deducted, the remainder is 3 (hundreds,) to which we join the next period, 25, making the dividend 325. These 3 (tens=30) in the quotient, or root, it will be recollected, are the length of one side of a square field which contains 9 (hundred) square rods, which are contained in the square figure A 30 rods. 5 rods. (30 X 30=900,) and 900 square rods deducted
from 1225 square rods leaves 325 square rod 30
to be added to the square figure A. D
Now to dispose of the remaining 325 rods so 150 25
as to retain the square form of the figure A, it is evident that we must make the addition on
two sides : the length of each is 30 rods, and А
30+30=60 rods, or which is the same thing,
double the root already found: then, 3 (tens) 30
makes 6 (tens) or 60 for a divisor ; then if we 30
30 divide 325 by 60, or, which is the same thing, 900 sq. rods.
neglect the cipher in the divisor and divide 6 150
(tens) into 32 (tens,) it shows that the breadth
of the addition must be 5 rods, which is the 30 rods.
next figure in the root ; then if we examinė the figure, we shall find that it is not yet complete, for there yet remains a small square in the corner D, each side of which is 5 rods,=to the last quotient figure, This quotient figure, 5, we must add to the divisor 60, (by the Rule) making 65 the whole divisor. Or, which is the same thing, we place the figure 5 at the right
hand of the 6 (tens,) making 65 the whole divisor, which multiplied by the quotient figure 5 gives the whole number of square rods in the addition around the sides of the square, A : 65 X5=325 square rods, which being deducted from the dividend, 325, leaves 00.
Hence we find that the square root of 1225 is 35, which is the length of one side of the field.
2 Proof:- This question may be proved by Involution ; thus, 35, or 35 X 35=1225 : or by adding the several parts of the figure together ;
Thus, A contains 900 square rods.
1225 square rods as before. 2. What is the square root of 55696 ?
55696 3. What is the square root of 6440,0625 ? 6440,0625(80,25 . Ans.
6440,0625 4. What is the square root of 138384 ? Ans. 372. 5. What is the square root of 1648,36 ? Ans. 40,6. 6. What is the square root of 6625476? Ans. 2574. 7. What is the square root of 488,631025 ? Ans. 22,105. 8. What is the square root of ,002304? Ans. ,048. 9. What is the square root of 30138696025 ?
Ans. 173605. 10. What is the square root of 1355 ?-In this example, after bringing down all the figures there is a remainder, to which we annex a period of ciphers and continue the operation to decimals; and there is still a remainder, to
which we annex another period of ciphers, and proceed as before.-And in this way we may continue the operation to any assigned degree of exactness. But we can never obtain the precise root, for the right hand figure in the dividend will always be a cipher; but the last figure in each divisor is the same as the last figure in the root, and no one of the nine digits, multiplied into itself, will produce a number ending with a cipher. Consequently, there will always be a remainder, whatever may be the quotient figure.
Ans. 36,81+ 11. What is the square root of 45? Ans. 6,708+. 12. What is the square root of 2972 ? Ans. 54,516+. 13. What is the square root of 1546,8 ? Ans. 39,3293+.
TO EXTRACT THE SQUARE ROOT OF VULGAR FRACTIONS.
Reduce the fraction to its lowest terms, and extract thë root of the numerator for a new numerator, and the root of the denominator for a new denominator.
If it be a mixed number, reduce it to an improper frac. tion; or if it be a surd, reduce it to a decimal; and then extract its root.
14. What is the square root of 197?– Thus, 193=48; then v 49=7, the numerator; and v 64=8; the denominator.
Ans. 1 15. What is the square root of 125?
Ans. 16. What is the square root of 225..? 17. What is the square root of 187 ?
Ans. 4. 18. What is the square root of 162 16? Ans. 12.
19. What is the square root of 26 ? 20. What is the square root of 11 ?
Ans. ,866+: Ans. ,9574+.
APPLICATION AND USE OF THE SQUARE ROOT. 1. A certain square pavement contains 25600 square stones of equal size ; how many are contained in one of its šides ?
2. A General has an army of 5625 men; how many must be placed in rank and file to form them into a square ?
5625=75, Ans. 3. How many trees in each of the rows of a square orchard that contains 5184 ?
V5184=72, Ans. 4. There is a circle whose area, or superficial contents, is 2025 feet; what is the length of the side of a square containing an equal number of feet ? Ans. 45 feet.
PROBLEM 11.—The square root of the product of any two numbers is a mean proportional between thosë numbers. 1. What is the mean proportional between 19 and 76 ?
176x19=38, Ans, 2. A certain field is 84 rods long and 21 rods wide ; what is the length of a square field which shall contain an equal number of rods?
184 x 21=42 rods, Ans. PROB. III.--The length of the side of a square being given to make another square which shall contain 2, 3, 4, &c. times as much.
Multiply the square of the given side by the given proportion, and extract the square root of the product.
1. The length of the side of a certain square garden is 12 rods; I demand the length of the side of another garden containing 4 times as much ? V12 x 12 x 4=24rds. Ans.
2. If the side of a square be 5 feet, what is the length of the side of another square which will contain 25 times as much ?
75X5 X25=25 feet, Ans. PROB. IV.—To place any number of soldiers so that the number in rank shall be double, triple, &c, the number in file, (or, to make a figure of any given number, so that the length shall be double, triple, &c. the breadth.)
Extract the square root of i, j, &c., of the given number, which will be the number in file, or (breadth ;) double, triple, &c. the number in file, (or breadth,) and that will be the number in rank, (or length.)