Exponents of Powers and Roots.

§ 293. An exponent or index is an integer annexed to a number to denote a power, or a fraction annexed to denote a root, of that number.

Thus, 52 denotes the 2d power or square of 5.

53 denotes the 3d power or cube

of 5.

16 denotes the square root of 16; 27 the cube root of

27, &c.

In these expressions, 2, 3, and are exponents. An exponent is always set on the right of the number, and a little elevated, as in the examples.

A root is also denoted by the radical sign, with an integral exponent or index.

Thus, 9 denotes the square root of 9; 3/27, the cube root of 27; 4/100, the 4th root of 100; and so on.

INVOLUTION.

$ 294. Involution consists in raising a given number to any required power. Thus, in finding the square of 25, we perform an involution on 25.

§ 295. A higher power of a given number may be found, most readily, by multiplying together two or more known powers, the sum of whose exponents is equal to the exponent of the required power.

Thus, the square of a number multiplied into itself, produces the 4th power of that number. 32×32=3×3×3×3=31. The square the cube, produces the 5th power. 32×33=3 X3X3X3X3=35; and so on.

EVOLUTION.

§ 296. Evolution consists in extracting any required root of a given number regarded as the corresponding power of the root to be found.

Thus, in extracting the square root of 625, we perform an evolution on 625.

The extraction of the square root of a number, consists in finding one of the two equal factors which, multiplied together, produce the given number.

Thus we find 25 to be the 625, because 25X25=625.

Extraction of the Square Root

The following principles are involved in the Rule to be given for extracting the square root.

§ 297. I. The square of any number has, at most, only twice as many figures, and, at least, only one less than twice as many, as the number itself.

Thus, 92-81; 992-9801; 9992=998001; in which examples, the squares 81, &c., have only twice as many figures as the numbers 9, &c., and these numbers are the largest that can be expressed by the same number of figures.

Again; 102=100; 1002=10000; 10002 1000000; in which examples, the squares 100, &c. have only one less than twice as many figures as the numbers 10, &c., and these numbers are the smallest that can be expressed by the same number of figures.

From the preceding, it follows, that,

§ 298. II. A number has two figures for each figure in its square root, excepting the left hand one-for which it has one, or two figures, according as said left hand figure produces one or two figures in squaring the root.

§ 299. III. If a number be divided into any two parts, the square of the number will be equal to

the square of the 1st part + twice the 1st 2d the square of the 2d part.

For example, 16=10+6, and the square of 16= the square of (10+6).

10+6

100+120+36= the square of (10+6)=162.

In squaring 10+6, that is, in multiplying it by itself, we first multiply 10+6 by 6; this gives 60+-36 We then multiply 10+6 by 10; which gives 100+60. The sum of the two products is 100+120+36=162.

This square is composed of 102-100, twice 10×6=120, and 62=36. Thus we see that the square of the number 16 is equal to the square of the first part 10, + twice the product of the two parts 10 and 6, + the square of the second part 6.

RULE LVII.

$ 300. To extract the Square Root of a given number.

1. Separate the given number into periods of two figures each, from right to left; observing that the last period may sometimes have but one figure.

2. From the left hand period subtract the greatest square number it contains, and set the root of said square for the first figure of the root required.

3. To the remainder affix the next period for a dividend. Divide this dividend, exclusive of its right hand figure, by twice the root already found, and annex the quotient figure to both the root and the divisor.

4. Multiply the divisor thus increased, by the quotient figure; subtract the product from the dividend; to the remainder affix the next period; divide by twice the root already found; and so on, till the operation is completed.

The left hand period is 5, and the greatest square number it contains is 4, the root of which is 2. Subtracting, and to the remainder 1 affixing the next period, we have for a dividend 129. Excluding its right hand figure 9, we divide 12 by 4, which is twice the root 2 already found, and annex the quotient figure 3 to both the root and the divisor. Multiplying the divisor 43 by 3, completes the operation.

The left hand period 36 being a square number, we set its root 6 for the first figure in the root required; and set down the next period 84, for a dividend.

Excluding its right hand figure 4, and dividing 8 by 12, which is twice the root 6 already found, the quotient is 0, which we annex to the root 6 and divisor 12.

To 84 we affix the next period 49, and the dividend is 8449. Excluding its right hand figure, and dividing by 120, which is twice the root 60 already found, the quotient is 7. Annexing this to the root and divisor, and multiplying, the operation is completed.

The given number is separated into periods to determine the number of figures in the root; and also whether one or two figures on the left correspond to the first figure in the root; (§ 298). In the first example,

The first figure 2 in the root is 2 tens 20, and its square is 400; which, subtracted, leaves the remainder 129.

The given number 529

root, twice 2 tens

of the units. (§ 299).

the

the square of 2 tens, the 1st part of its units or 2d part of the root, the square

And since the square of 2 tens has been subtracted, the remainder 129 twice 2 tens the units + the square of the units.

In taking 4, twice 2, for a divisor, we omitted the 0 on the right of 2 tens. Omitting, therefore, the 9 in the corresponding place of the dividend 129, we say 4 in 12, 3 times.

The quotient 3 annexed to the 4 in the divisor, makes the 4 become 4 tens; and the divisor 4 tens +3, multiplied by 3,=4 tens ×3+32; or twice 2 tens the units in the root, the square of the units, =129. Hence the second figure in the root is correctly found by the Rule.

When the given number contains three or more periods, the first and second figures of the root having been found, these two figures together may be regarded as the 1st part of the entire root, and the remainder of the root as the 2d part. Three figures of the root having been found, these may together be regarded as the 1st part of the root, and the remainder as the 2d part; and so on.

Under these views, the principle on which depends the method of forming the divisors, (§ 299), becomes applicable when there are three or more, as well as when there are but two figures in the root.

EXERCISES.

1. Find the square root of 784, and of 11236.

Ans. 28; and 106.

2. Find the square root of 2025, and of 38809.

Ans. 45; and 197.

3. Find the square root of 7396, and of 75076.

Ans. 86; and 274.

4. Find the square root of 22801, and of 473344.

Ans. 151; and 688.

5. Find the square root of 36100, and of 904401.

Ans. 190; and 951.

6. Find the square root of, and of 1158.

5041

Ans. 1; and 4.

7. Find the square root of §591, and of 10000.

39204

Ans. ; and 188.

8. Find the square root of $88, and of 11881.

Ans. ; and 18.

1225

9. Find the square root of 18, and of 15.

32 and 3.

Ans. ;

10. Find the square root of

t, and of 94%

97

Ans. ; and 0.

Square Root of Decimals, Imperfect Squares, &c.

§ 301. In extracting the square root, an integer is separated into periods from right to left.

A decimal must be separated into periods of two figures each, from the decimal point towards the right, and a 0 annexed when necessary to complete the last period.

The number of decimal figures in the root, will be equal to the number of periods in the given decimal.

In finding the root of an imperfect square, (§ 292), a period of Os may be annexed to the last remainder, and the operation continued in this manner to any required exactness ;-observing that each period thus annexed must be counted as a decimal period belonging to the given number.

A fraction will be an imperfect square, if either of its terms is an imperfect square. Its root in such case will be found, most readily, by extracting the root of its equivalent decimal.

A vulgar fraction annexed to an integer may be reduced to a decimal, and the root of the mixed number be then extracted.