By examining the several powers of 3 in the examples given, we see that the index of each power is equal to the number of times 3 is used as a factor in the multiplications producing the power, and that the number of times the number is multiplied into itself is one less than the power denoted by the index. Hence the

RULE. - Multiply the given number by itself, as many times less 1, as there are units in the exponent of the required power.

NOTE 1. A fraction may be raised to any power, by involving its terms Thus, the second power of is X }

=

NOTE 2. A mixed number may be either reduced to an improper fraction, or the fractional part reduced to a decimal, and then raised to the required power.

278. To raise a number to any required power without producing all the intermediate powers.

Ex. 1. What is the 8th power of 4?

Ans. 65536.

We raise the 4 to the 2d and to the 3d power, and write above each power its exponent. We then add the exponent 3 to itself, and, increasing the sum by the exponent 2, obtain 8, a number equal to the power required. We next multiply 64, the power belonging to the exponent 3, into itself, and this product by 16, the power belonging to the exponent 2, and obtain 65536 for the 8th power.

RULE. Raise the given number to any convenient number of powers, and write above each of the respective powers its exponent. Then add

277. The rule for raising a number to any required power? How may a common fraction be raised to a required power? How a mixed number? 278. What are the numbers placed over the several powers of 4 called, and what do they denote ?

together such exponents as will make a number equal to the required power, repeating any one when it is more convenient, and the product of the powers belonging to these exponents will be the required answer.

EXAMPLES FOR PRACTICE.

2. What is the 7th power of 5?

3. What is the 9th power of 6?
4. What is the 12 power of 7?
5. What is the 8th power of 8?
6. What is the 20th power of 4?
7. What is the 30th power of 3?

Ans. 78125.

Ans. 10077696. Ans. 13841287201.

Ans. 16777216.

Ans. 1099511627776. Ans. 205891132094649.

EVOLUTION.

279. Evolution is the process of finding the root of a given power. It is the reverse of Involution.

A Root of a power is a number which, being multiplied into itself a certain number of times, will produce the given power. Thus 4 is the second or square root of 16, because 4 X 4 = 16 and 3 is the third or cube root of 27, because 3 × 3 × 3

= = 27. Roots take the name of the corresponding power, thus ; The Second, or Square Root, is that of a second power. The Third, or Cube Root, is that of a third power. The Fourth, or Biquadrate Root, is that of a fourth power. Rational Roots are those roots which can be exactly found. Surd Roots are those which cannot be exactly found, but approximate towards true roots.

Numbers that have exact roots are called perfect powers, and all other numbers are called imperfect powers.

278. The rule for involving a number without producing all the intermediate powers?-279. What is Evolution? What is a root? From what does the root take its name? What are rational roots? Surd roots?

Roots are denoted by writing the character, called the radical sign, before the power, with the index of the root over it, or by a fractional index or exponent; in case, however, of the second or square root, the index 2 is omitted. The third or cube root of 27 is expressed thus,

root of 25 is expressed thus,

27, or 27; the second or square 25, or 25; and the square of the

cube root of 27, or the cube root of the square of 27, is expressed thus, 273.

EXTRACTION OF THE SQUARE ROOT.

280. The Square Root, or the root of a second power, is so called because the square or second power of any number represents the contents of a square surface, of which the root is the length of one side.

281. To extract the square root of a number is to find such a factor as, when multiplied by itself, will produce the given number, or it is to resolve the number into two equal factors.

Roots of the first ten integers and their squares are:—

1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

It will be observed that the second power or square of each of the numbers contains twice as many figures as the root, or twice as many wanting one. Hence, to ascertain the number of figures in the square root of a given number,

Beginning at the right, point it off into as many periods as possible, of two figures each; and there will be as many figures in the root as there are periods.

NOTE. When the given number contains an odd number of figures, the period at the left can contain but one figure.

Ex. 1. I wish to arrange 625 tiles, each of which is 1 foot square, into a square pavement; what will be the length of one of the sides? Ans. 25 feet.

OPERATION.

625 (25, Ans.

4

45) 225 225

279. How are roots denoted?

We must extract the square root of 625 to obtain one side of the pavement, in feet. (Art. 280.)

Beginning at the right hand, we point off the number into periods, by placing a point over the right-hand figure of each period; and then find the greatest square

What is said of the index 2?-280. What is meant by the square root, and why is it so called? 281. What is meant by extracting the square root? How do you ascertain the number of figures in the square root of any number?

number in the left-hand period, 6 (hundreds) to be 4 (hundreds), and that its root is 2, which we write in the quotient. As this 2 is in the place of tens, its value is 20, and represents the side of a square, the area or superficial contents of which are 400 square feet, as seen in Fig. 1.

The width of the additions, makes the

25 feet.

Fig. 2.
25 feet.

20

E TOO

D

20

F

20

25

We now subtract 400 feet from 625 feet, and have 225 feet remaining, which must be added on two sides of Fig. 1, in order that it may remain a square. We therefore double the root 2 (tens) or 20, one side of the square, to obtain the length of the two sides to be enlarged, making 40 feet; and then inquire how many times 40, as a divisor, is contained in the dividend 225, and find it to be 5 times. This 5 we write in the quotient or root, and also on the right of the divisor, and it represents the width of the additions E and F to the square, as seen in Fig. 2.

additions multiplied by 40, the length of the two contents of the two additions E and F to be

25 feet.

200 square feet, or 100 feet for each. The space G now remains to be filled, to complete the square, each side of which is 5 feet, or equal to the width of E and F. We square 5, and have the contents of the last addition, G, equal to 25 square feet. It is on account of this last addition that the last figure of the root is placed in the divisor; for we thus obtain 45 feet for the length of all the additions made, which, being multiplied by the width (5ft.), the last figure in the root, the product, 225 square feet, will be the contents of the three additions, E, F, and G, and equal to the feet remaining after we had found the first square. Hence, we obtain 25 feet for the length of one side of the pavement, since 25 X 25625, the number of tiles to be arranged, and equal to the sum of the several parts of Fig. 2; thus, 400 + 100 + 100+ 25

20

400

100

25 feet.

= 625.

This illustration and explanation is founded upon the principle, That

281. What is first done after dividing the number into periods? What part of Fig. 1 does this greatest square number represent? What place does the figure of the root occupy, and what part of the figure does it represent? Why do you double the root for a divisor? What part of Fig. 2 does the divisor represent? What part does the last figure of the root represent? Why do you multiply the divisor by the last figure of the root? What parts of the figure does the product represent? Why do you square the last figure of the root? What part of the figure does this square represent?. What other way of finding the contents of the additions without multiplying the parts separately by the width ?

the square of the sum of two numbers is equal to the squares of the num bers, plus twice their product. Thus, 25 being equal to 20+5, its square is equal to the squares of 20 and of 5, plus twice the product of 20 and 5, or to 400 + 2 X 20 X 5+25

RULE.

== 625.

Point off the given number into as many periods as possible of two figures each, by putting a point over the place of units, another over the place of hundreds, and so on.

Find the greatest square number in the left-hand period, writing the root of it at the right-hand of the given number, after the manner of a quotient in division, for the first figure of the root. Subtract this square

number from the first period, and to the remainder bring down the next period for a dividend.

Double the root already found for a divisor, and find how often the divisor is contained in the dividend, exclusive of the right-hand figure, and annex the result to the root for the second figure of it, and likewise to the divisor.* Multiply the divisor with the figure last annexed by the figure annexed to the root, and subtract the product from the dividend. To the remainder bring down the next period for a new dividend.

Double the root already found for a new divisor, and continue the operation as before, till all the periods have been brought down.

NOTE 1. The left-hand period may contain but one figure. (Art. 281, Note.)

2. If the dividend does not contain the divisor, a cipher must be placed in the root, and also at the right of the divisor; then, after bringing down the next period, this last divisor must be used as the divisor of the new dividend.

3. When there is a remainder after extracting the root of a number, periods of ciphers may be annexed, and the figures of the root thus obtained will be decimals.

4. If the given number is a decimal, or a whole number and a decimal, the root is extracted in the same manner as in whole numbers, except, in pointing off the decimals, either alone or in connection with the whole number, we place a point over every second figure toward the right, from the separatrix, and fill the last period, if incomplete, with a cipher.

5. The square root of any number ending with 2, 3, 7, or 8, cannot be exactly found.

EXAMPLES FOR PRACTICE.

2. What is the square root of 148996?

The figure of the root must generally be diminished by one or two units, on account of the deficiency in enlarging the square.

281. The rule for extracting the square root? What is to be done if the dividend does not contain the divisor? What must be done if there is a remainder after extracting the root? What do you do if the given number is a decimal? Of what numbers can the exact square root not be found?