TO EXTRACT THE SQUARE ROOT OF VULGAR FRACTIONS.

RULE.

Reduce the fraction to its lowest terms for this and alt other roots; then

1. Extract the root of the numerator for a new numeratur, and the root of the denominator, for a new denominator. 2. If the fraction be a surd, reduce it to a decimal, and extract its root.

PROBLEM I.-A certain general has an army of 5184 men; how many must he place in rank and file, to form them into a square?

RULE.-Extract the square root of the given number. √5184-72 Ans. PROB. II. A certain square pavement contains 20736 square stones, all of the same size; I demand how many are contained in one of its sides? 20736=144 Ans. PROB. III. To find a mean proportional between two numbers.

RULE.-Multiply the given numbers together and extract the square root of the product.

EXAMPLES.

What is the mean proportional between 18 and 72? 72 × 18=1296, and ✓1296=36 Ans. PROP. IV. To form any body of soldiers so that they may Le double, triple &c. as many in rank as in file.

RULE.-Extract the square root of 1-2, 1-3, &c. of the given number of men, and that will be the number of men in file,which double, triple, &c. and the product will be the number in rank.

EXAMPLES.

Let 13122 men be so formed, as that the number in rank may be double the number in file.

13122÷2=6501, and √6561=81 in file, and 81×2 -162 in rank.

PROP. V. Admit 10 hhds. of water are discharged through a leaden pipe of 24 inches in diameter, in a certain time; I demand what the diameter of another pipe must be to discharge four times as much water in the same

time.

RULE.-Square the given diameter, and multiply said square by the given proportion, and the square root of the product is the answer.

21-2,5, and 2,5×2,5=6,25 square.

4 given proportion.

'25,00-5 inch. diam. Ars.

PROB. VI. The sum of any two numbers, and their proJucts being given, to find each number.

RULE. From the square of their sum, subtract 4 times their product, and extract the square root of the remainder, which will be the difference of the two numbers; then half the said difference added to half the sum, gives the greater of the two numbers, and the said half difference subtracted from the half sum, gives the lesser number.

EXAMPLES.

The sum of two numbers is 43, and their product is 412; what are those two numbers?

The sum of the numb. 43 × 43-1849 square of do. The product of do. 442× 4=1763 4 times the pro.

Then to the sum of 21,5

4 the dif

Answers.

EXTRACTION OF THE CUBE ROOT. A cube is any number multiplied by its square.

To extract the cube root, is to find a number, which, being multiplied into its square, shall produce the given number.

RULE.

1. Separate the given number into periods of three figures each, by putting a point over the unit figure, and every third figure from the place of units to the left, and if there be decimals, to the right.

2. Find the greatest cube in the left hand period, and place its root in the quotient.

3. Subtract the cube thus found, from the said period. and to the remainder bring down the next period, calling this the dividend.

4. Multiply the square of the quotien, by 300, calling it the divisor.

5. Seek how often the divisor may be had in the divi dend, and place the result in the quotient; then multiply the divisor by this last quotient figure, placing the product under the dividend.

6. Multiply the former quotient figure, or figures, by the square of the last quotient figure, and that product by 30, and place the product under the last; then under these two products place the cube of the last quotient figure, and add them together, calling their sum the subtrahend.

7. Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend; with which proceed in the same manner, till the whole be finished.

NOTE. If the subtrahend (found by the foregoing rule) happens to be greater than the dividend, and cor equently cannot be subtracted therefrom, you must make the last quotient figure one less; with which find a new subtrahend, (by the rule foregoing,) and so on until you can subtract the subtrahend from the dividend.

EXAMPLES.

1. Required the cube root of 18399,744.

18399,744(26,4 Root. Ans.

8

2×2=4×300=1200)10399 first dividend.

7200

6×6=36×2=72×30=2160

6×6×6 216

9576 1st subtrahend.

811200

26:<26=676 × 300—202800)823744 2d dividend.

4×4=16×26=416 × 30= 12480

4×4×4

64

823744 2d subtrahend

NOTE. The foregoing example gives a perfect root; and if, when all the periods are exhausted, there happens to be a remainder, you may annex periods of ciphers, and cortinue the operation as far as you think it necessary.

RULE. 1. Find by trial, a cube near to the given number, and call is the supposed cube.

2. Then, as twice the supposed cubo, added to the given number, i to twice the given number added to the supposed cube, so is the root of the supposed cube, to the true root, or an approximation to it.

3. By taking the cube of the root thus found, for the supposed cube, and repeating the operation, the root will be had to a greater degree of exactness,

EXAMPLES.

1. Let it be required to extract the cube root of 2. Assume 1,3 as the root of the nearest cube; then-1,3 x 1,3×1,3=2,197=supposed cube.

Then, 2,197

2,000 given number.

As 6,394

1,3 : 1,2599 root,

which is true to the last place of decimals; but might by repeating the operation, be brought to greater exactness. 2. What is the cube root of 584,277056 ?

Ans. 3.30.