270. If the square root of a number have decimal places, the number itself will have twice as many.

=

Thus, if 0.11 be the square root of some number, the number will be (0.11)2= 0.11 0.11 0.0121. Hence, if a given square number contain a decimal, and if it be divided into periods of two figures each, by beginning at the decimal-point and marking toward the left for the integral number, and toward the right for the decimal, the number of periods to the left of the decimal-point will show the number of integral places in the root, and the number of periods to the right will show the number of decimal places in the root. Extract the square root of 52.2729.

52.27 29 (7.23

49 142)3 27 284

1443) 43 29

43 29

It will be seen from the periods that the root will have one integral and two decimal places.

271. If a number is not a perfect square, ciphers may be annexed, and an approximate value of the root found. Extract to six places of decimals the square root of 19. 19 00 00 00 (4.358899

272. The square root of a common fraction is found by extracting the square roots of the numerator and denomi

nator. But, when the denominator is not a perfect square, it is best to reduce the fraction to a decimal and then extract the root.

The side of a square is found by extracting the square root of its area.

17. A rectangle is 972 yds. long and 432 yds. wide. Find the side of a square which has the same area as the rectangle.

18. Find in yards the length of

the side of a square field

containing 27 A. 12 sq. rds.
1 sq. yd.

In a right triangle, the square on the hypotenuse (AC) is equal to the sum of the squares on the two legs.

A

C

19. Base 39, perpendicular = 52; find hypotenuse. 20. Base 35, hypotenuse 91; find perpendicular. 21. Perpendicular 72, hypotenuse 75; find base. 22. A cord 287 ft. long is stretched from the top of a flagpole 63 ft. high; find the distance of the end in contact with the ground from the base of the pole.

The length of the diagonal of a room is the square root of the sum of the squares of the length, breadth, and height. 23. Find the diagonal of a room 28 ft. long, 21 ft. wide, and 12 ft. high.

24. Find the diagonal of a hall 50 ft. long, 30 ft. wide, and 15 ft. high.

CUBE ROOT.

273. The cube of a number is the product of three factors, each equal to the number.

are

The cubes of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

274. The cube root of a number is one of the three equal factors of the number.

are

Thus the cube roots of 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

275. The cube root of a number is indicated by V, or by the exponent written above and to the right of the

number.

Thus, 343, or 3433, means the cube root of 343,

276. Since 35 = 30+5, the cube of 35 may be obtained

Hence the cube of any number composed of tens and units contains four parts:

I. The cube of the tens.

II. Three times the product of the square of the tens by the units.

III. Three times the product of the tens by the square of the units.

IV. The cube of the units.

277. In extracting the cube root of a number, the first step is to mark it off in periods.

Since 1

13, 1000 = 103, 1,000,000 = 1003, and so on, it follows that the cube root of any number between 1 and 1000, that is, of any number that has one, two, or three figures, is a number of one figure; and that the cube root of any number between 1000 and 1,000,000, that is, of any number that has four, five, or six figures, is a number of two figures, and so on.

If, therefore, an integral number be divided into periods of three figures each, from right to left, the number of figures in the root will be equal to the number of periods. The last period may consist of one, two, or three figures.

Extract the cube root of 42875.

27 is 3. Hence 3 is the tens' figure of the root.

The remainder, 15875, resulting from subtracting the cube of the tens, will contain three times the product of the square of the tens by the units three times the product of the tens by the square of the units

the cube of the units.

Each of these three parts contains the units' number as a factor.

Hence the 15875 consists of two factors, one of which is the units' number of the root; and the other factor is three times the square of the tens + three times the product of the tens by the square of the units the square of the units. The larger part of this second factor is three times the square of the tens.

And, if the 158 hundreds of the remainder be divided by the 3 × 302 = 27 hundreds, the quotient will be the units' number of the

root.

The second factor can now be completed by adding to the 2700 3 × (30 × 5) = 450 and 52 = 25.

278. The same method will apply to numbers of more than two periods, by considering the part of the root already found as so many tens with respect to the next figure of the root.

Extract the cube root of 57512456.

279. If the cube root of a number have decimal places, the number itself will have three times as many.

=

Thus, if 0.11 be the cube root of a number, the number is 0.11 × 0.11 0.11 0.001331. Hence, if a given number contain a decimal, and if it be divided into periods of three figures each, by beginning at the decimal-point and marking toward the left for the integral number and toward the right for the decimal, the number of periods toward the left from the decimal-point will show the number of integral places in the root, and the number of periods toward the right will show the number of decimal places in the root.