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And, if taken 4 times as a factor, the product is the 4th power; if 5 times, the product is the 5th power, and so on.

Hence, the different powers derive their name from the number of times the root is taken as a factor.

REM. The given number is called the root, the different powers of the number being derived from it.

ART. 278. The second power of a the square; the third power the cube. derived thus:

ILLUSTRATIONS. is the line itself.

number is called These terms are

1. Take a line, say 3 feet long, its first power

2. If 3 feet be multiplied by itself, the product (Art. 87) will be 3X3 9 square feet. (See diagram of 3 feet square, page 91.) But 3X3 9 is the second power of 3; hence, the 2d power is called the square.

3. If each side of a cube is 3 feet, the cube (Art. 92) contains 3×3×3=27 cubic ft. (See diagram, p. 94.) But 3 × 3 × 3 = 27, is the third power of 3; hence, the 3d power is called the cube.

ART. 279. The number denoting the power to which the root is to be raised, is the index or exponent of the power. It is placed on the right, a little higher than the Thus,

root.

212, the 1st power of 2.

22=2×2=4, the 2d power or square of 2.
23=2×2×28, the 3d power, or cube of 2.
24=2X2X2X2=16, the 4th power of 2, &c.

To find the second power of 2, use it as a factor twice; thus, 2×2=4. To find the third power of 2, use it three times; thus, 2×2×2=8, and so on.

REVIEW.-277. What is involution? What is a power? What is the root, or 1st power? The 2d power? The 3d? The 4th ? 277. From what do the different powers derive their name is the given number called the root? 278. What is the second number called? What the 3d? How are these terms derived?

is the index of a power? How find the 2d power of 2? The 3d?

REM. Why

power of a 279. What

The 4th?

Rule for Involution.-Multiply the number by itself, till it is used as a factor as many times as there are units in the index of the power.

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5. What is the fourth power of 2? fifth power of 2? fourth power of 3?

SLATE EXERCISES.

6. What is the 2d power or sq. of 65? Ans. 4225.

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XXIV. EVOLUTION.

ART. 280. EVOLUTION, or the extraction of roots, is the process of resolving numbers into equal factors.

When a number is resolved into equal factors, each factor is a ROOT of the number.

Hence, a root is a factor which, multiplied by itself a certain number of times, will produce the given number.

One of the two equal factors of a number, is the second root, or square root of that number. Thus,

9-3X3; three being the square root of 9.

One of the three equal factors of a number, is the third, or cube root of that number. Thus,

8=2X2X2; two being the cube root of 8.

Also, one of the four, five, &c., equal factors of a number is the fourth, fifth, &c., root of that number.

Hence, the name of the root shows the number of equal factors into which the given number is resolved. Thus. The square root of 25 is 5, as 5 X 5 = 25.

The cube root of 27 is 3, as 3X 3X3:

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27.

The fourth root of 16 is 2, as 2X2 X2 X2=16, &c.

Evolution is the reverse of Involution. In Involution, the root is given to find the power; in Evolution, the power, to find the root. When one number is a power of another, the latter is a root of the former: thus, 8 is the cube of 2, and 2 is the cube root of 8.

ART. 281. ROOTS ARE DENOTED IN TWO WAYS:

1st. By

called the radical sign, placed before the number. 2d. By a fractional index placed on the right of the number. Thus, √ 4, or 4, denotes the square root of 4.

And, 3/27, or 273, the cube root of 27.

NOTE. The figure over the radical sign, denotes the name of the root. When the sign has no figure over it, 2 is understood; thus, 2/25 and 1/25, each denotes the square root of 25. The denominator of the fractional index denotes the name of the root.

ART. 282. Any number whose exact root can be obtained, is a PERFECT POWER: as, 4, 9, 16, &c.

REVIEW. 279. What is the rule for involution? 280. What is evolution? What is a root of a number? What the second or square root? What the third, or cube root? What does the name of the root show? Give examples. REM. Why is evolution the reverse of involution? 281. How are roots denoted? Give examples. NOTE. What does the figure over the radical sign denote? What the denominator of the fractional index? 282. What is a perfect power? Give examples.

ART. 283. Every root, and every power of 1, is 1.
Thus, 1, 3/1, 4/1, and (1)2, (1)3, (1)4, each=1.

EXTRACTION OF THE SQUARE ROOT.

ART. 284. To extract the square root of a number, is to resolve it into Two equal factors (Art. 280); or, to find a number, which, multiplied by itself, will produce the given number. 4 in. long.

The extraction of the square root (Art. 278) is also the method of finding the number of units in the side of a square, when its superficial contents are known; or, knowing the superficial contents, it shows how to arrange them so as to form the largest square possible.

EXAMPLE.

--

4 X 4 = 16.

4 in. wide.

-What is the side of a square board which contains 16 square inches?

SOLUTION. Since 164 X 4, each side is 4 inches.

ART. 285. THE FIRST TEN NUMBERS AND THEIR SQUARES ARE

4,

5,

Numbers. 1, 2, 3,
6, 7, 8, 9, 10.
Squares. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
The numbers in the 1st line are the square roots of those in the 2d.

Since the sq. root of 1 is 1, and of 100 is 10, the sq. root of any number less than 100 consists of one figure. That is, the square root of a number of fewer than three figures, must consist of only one figure.

Again, take the numbers 10, 20, 30, 40, &c., to 100.

Their squares are 100, 400, 900, 1600, &c., to 10000.

Since the square root of 100 is 10, and of 10000 is 100, the square root of any number greater than 100, and less than 10000, will consist of two figures.

REV. 283. What are the different roots and is it to extract the square root of a number?

powers of 1? 284. What What else may it be con

sidered? What is the side of a square containing 9 sq. in.? 25 sq. in.? 285. What the rule for pointing? Why?

The square root of a number of more than two figures and fewer than five, must consist of two figures.

Also, the square root of a number of more than four figures and less than seven, must consist of three figures.

Hence, if a dot (.) be placed over every alternate figure, beginning with units, the number of dots will be the number of figures in the root. This is the RULE FOR POINTING.

OPERATION.

ART. 286. 1. Extract the square root of 256; or, what is the same, arrange 256 sq. in. in the form of a square. SOL. To ascertain the number of figures in the root, begin at the unit's place, and place a dot over each alternate figure. This shows that the root consists of two figures.

256(10+6

100 10156

2156

20

6

26

=16

Ans.

Next find that the largest square in 2 (hundred) is 1 (hundred), the sq. root of which is 1 (ten), which put on the right, as in writing the quotient in division. Subtract the 100 from the given number, and 156 remain. While solving this example by figures, attend to arranging the squares.

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Without the diagram, it would be difficult to tell what operation to perform on the 156 to obtain the other figure of the root.

By examining the figure A, it is obvious, that to increase it, and at the same time preserve it a square, both length and breadth gt be increased equally; and since each side is 10in. long, it will

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