231. EXAMPLE.-Divide 820 rd. 4 yd. 2 ft. by 112.

EXPLANATION. The first quotient is 7 rods with 36 rods. remaining. 5.5 × 36 = 198 yards; 198 yards + 4 yards = 202 yards; 202 yards 1121 yard and 90 yards remaining. 90 x 3 270 feet; 270 feet +2 feet 272 feet; 272 feet 1122 feet and 48 feet remaining; 48 × 12 = 576 inches; 576 inches 112 5.143 inches, nearly. Ans.

=

The preceding example is solved by long division, because the numbers are too large to deal with mentally. Instead of expressing the last result as a decimal, it might have. been expressed as a common fraction. Thus, 576 ÷ 112 = 554 inches. The chief advantage of using a common fraction is that if the quotient be multiplied by the divisor, the result will always be the same as the original dividend. 232. Rule 34.-Find how many times the divisor is contained in the first or highest denomination of the dividend. Reduce the remainder (if any) to the next lower denomination, and add to it the number in the given dividend expressing that denomination. Divide this new dividend by the divisor. The quotient will be the next denomination in the quotient required. Continue in this manner until the lowest denomination is reached. The successive quotients will constitute the entire quotient.

EXAMPLES FOR PRACTICE.

233.

Divide

(a) 376 mi. 276 rd. by 22; (b) 1,137 bu. 3 pk. 4 qt. 1 pt. by 10; (c) 84 cwt. 48 lb. 49 oz. by 16; (d) 78 sq. yd. 18 sq. ft. 41 sq. in. by 18; (e) 148 mi. 64 rd. 24 yd. by 12; (ƒ) 100 tons 16 cwt. 18 Ib. 11 oz. by 15; (g) 36 lb. 18 oz. 18 pwt. 14 gr. by 8; (h) 112 mi. 48 rd. by 100.

234. Involution is the process of multiplying a number by itself one or more times. The product obtained by multiplying a number by itself is called a power of that number.

Thus, the second power of 3 is 9. since 3 × 3 are 9.

The third power of 3 is 27, since 3 × 3 × 3 are 27.
The fifth power of 2 is 32, since 2 × 2 × 2 × 2 × 2 are 32.

235. An exponent is a small figure placed to the right and a little above a number to show to what power it is to be raised, or how many times the number is to be used as a factor, as the small figures "" and below:

3'3 x 3 = 9.

5

3' = 3 × 3 × 3 = 27.

2 = 2 × 2 × 2 × 2 × 2 = 32.

236. The root of a number is that number which, used the required number of times as a factor, produces the number. In the above cases, 3 is a root of 9, since 3 × 3 are 9. It is also a root of 27, since 3 × 3 × 3 are 27. Also, 2 is a root of 32, since 2 × 2 × 2 × 2 × 2 are 32.

237. The second power of a number is called its

square.

Thus, 5' is called the square of 5, or 5 squared, and its value is 5 X 5 = 25.

238. The third power of a number is called its cube. Thus, 5 is called the cube of 5, or 5 cubed, and its value is. 5 X 5 X 5 = 125.

To find any power of a number :

239. EXAMPLE.-What is the third power, or cube, of 35?

242.

SOLUTION.- (1)2=1XIX =

3 × 3 × 3
8X8X8

27 Ans.

EXAMPLE.—What is the third power, or cube, of § ?

243. Rule 35.-(a) To raise a whole number or a deci mal to any power, use it as a factor as many times as there are units in the exponent.

(b) To raise a fraction to any power, raise both the numer ator and denominator to the power indicated by the exponent.

EXAMPLES FOR PRACTICE.

244. Raise the following to the powers indicated:

(a) 852.

(b) (H)2.

(c) 6.52.

(d) 144.

245. Evolution is the reverse of involution. It is the process of finding the root of a number which is considered as a power.

246. The square root of a number is that number which, when used twice as a factor, produces the number. Thus, 2 is the square root of 4, since 2 x 2, or 2', = 4. 247. The cube root of a number is that number which, when used three times as a factor, produces the number. Thus, 3 is the cube root of 27, since 3 × 3 × 3, or 33, =27. 248. The radical sign, when placed before a number, indicates that some root of that number is to be found. 249. The index of the root is a small figure placed over and to the left of the radical sign, to show what root is to be found.

Thus, 100 denotes the square root of 100.

125 denotes the cube root of 125.

256 denotes the fourth root of 256, and so on.

250. When the square root is to be extracted, the index is generally omitted. Thus, 100 indicates the square root of 100. Also, 225 indicates the square root of 225.

SQUARE ROOT.

=

251. The largest number that can be written with one figure is 9, and 9' 81; the largest number that can be written with two figures is 99, and 99' 9,801; with three figures 999, and 999 998,001; with four figures 9,999, and 9,999 99,980,001, etc.

=

In each of the above it will be noticed that the square of the number contains just twice as many figures as the number.

In order to find the square root of a number, the first step is to find how many figures there will be in the root. This