283. To find any required power of a number.

What is the cube of 12?

MODEL OPERATION.

123=12 × 12 × 12,=1728, Ans.

FORM.-The cube of 12 equals 12 taken as a factor three times, or 1728.

1. What is the 3d power of 5? 2. What is the 4th power of 6? 3. What is the 7th power of 2? 4. What is the 3d power of 7? 5. What is the cube of 9?

6. What is the cube of 8? 7. What is the square of 11? 8. What is the 4th power of 9 ? 9. What is the 5th power of ? 10.*What is the 4th power of 31⁄2?

LESSON II.
EVOLUTION.

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284. A Root of any number is one of the equal factors which produce that number; thus, 2 is the cube root of 8 because 2X2X2=8.

285. Evolution is the method of finding the root of a given power or number, and is the reverse of Involution.

(a.) When the exact root of any number cannot be found, we can find what is called an approximate root; that is one of the equal factors whose product is very nearly equal to the number in question.

286. The Radical Sign is a character (√) that is placed before numbers to denote a root; thus, 81 indicates the square root of 81. To denote other roots than the square root, a small figure is used with the radical sign; thus, 27=the cube root of 27; √3125 the fifth root of 3125.

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*NOTE.-Reduce mixed numbers to improper fractions.

SQUARE ROOT.

287. The Square Root is one of the two equal factors that produce a number.

(a.) Since the square of tens will give units of no denomination below hundreds, and the square of hundreds will give units of no denomination below ten thousands, &c., the two right-hand figures of any number will contain no part of the square of the denominations of the root above units, &c. ; therefore if any number be divided into periods of two figures each, commencing at the right, the number of periods will equal the number of figures in the root.

What is the square root of 3969 ?

ILLUSTRATION.*-I have a square board containing 3969 square inches: what are its dimensions?

FORM.-1. The length of the board is equal to the breadth, and the product of the length by the breadth gives the area; hence the square root of 3969 sq. in., the area, will be the length of each side of the board.

*NOTE-The extraction of the square root of an abstract number may be illus. trated by the following example.

2. Separate the number into periods of two figures counting from the right (287., a.) thus 39'69, which shows that the root will contain two places.

3. By trial the length of the largest square in the left-hand period is 60 in., and the product of the length by the breadth is 3600 sq. in. (Fig. 1), which subtracted from 3969 sq. in. leave a remainder of 369 sq. in.

4. In order to preserve the form of a square, the remainder must be added to two of the sides. (Fig. 2.)

5. Since the length of one of the sides is 60 in., and the length of the other side is 60 in., the entire length must be twice 60 in., which are 120 inches.

6. Since there are 120 sq. in. in one row of the entire addition, the addition will be as many rows wide as 120 sq. in. is contained times in 369 sq. in., which is 3; and if the length of the entire addition is 120 sq. in., and the width is 3 in., the product will be 3 times 120 sq. in., which is 360 sq. in. (Fig. 2); 360 sq. in. subtracted from 369 sq. in. leave 9 sq. in.

7. To complete the square (Fig. 2), it will be necessary to add a small square 3 in. long and 3 in. widẹ (Fig. 3), containing 9 sq. in., which subtracted from 9 sq. in. leave no remainder.

Therefore, the board is 63. in. long and 63 in.wide.

What is the square root of 391 ?

MODEL OPERATION.

3'91(19.77+root.

Trial divisor, 1×20=20|291, dividend.
Complete divisor, 20+9=29 261

Trial diviзor, 19 × 20=380|30.00, dividend.

Complete divisor, 380+7=387 27.09

Trial divisor, 197 × 20=3940|2.9100, dividend.

Complete divisor, 3940+7=3947 2.7629

.1471, remainder.

For convenience, the following rule is adapted to the above contracted operation.

(a.) Rule.-I. Separate the number into periods of two places each counting toward the left from units' place.

II. Find the root of the left hand period; subtract its square from the period, and to the remainder annex the next period for a dividend.

III. Multiply the root found by 20, for a trial divisor; divide and take the quotient for the next figure in the root; also add the quotient to the trial divisor to complete the divisor.

IV. Multiply and subtract as in simple division, and to the remainder annex the next period for a new dividend.

V. Proceed in the same way until all the periods are used.

NOTES.-1. If the dividend does not contain the complete divisor, place a cipher in the root, and bring down the next period as a new dividend.

2. If the given number contains a decimal not having an equal number of decimal places, make them even by annexing ciphers before separating into periods; thus 341.672-3'41.67'20.

3. When the exact root of the terms of a common fraction cannot be found, reduce it to a decimal, and extract the root of the decimal

288. The Cube Root of any number is one of the three equal factors whose product equals that number; to extract the cube root of any number is to find that factor.

(a.) Since the cube of 10 is 1000, of 100 is 1,000,000, &c., it follows that the cube of 10 will give units of the denomination below thousands, &c. ; therefore, the three right-hand figures of any number will contain no part of the cube of the denominations above units, the six right-hand figures will contain no part of the cube above tens, &c.; hence, if any number be separated into periods of three figures commencing at the right, the number of periods will equal the number of figures in the root.

What is the cube root of 74088?

ILLUSTRATION.*-I have a cubical block containing 74088 cu. in.; what are its dimensions?

*NOTE.--The extraction of the cube root of an abstract number may be illustrated by the following example.