Thus, V/64 shows that the 3d root of 64 is to be taken; √81, the 4th root of 81; V15, the 1st root of 15, &c.

The index is usually omitted in case of the second or square root. Thus, √64 or 64 equally indicates the square root of 64.

The root of a number may also be indicated by a fractional exponent, placed on the right of the number. Thus 16 indicates the square root of 16; 81, the fourth root of 81.

12 denotes that the cube root of the square of 12 is to be taken.

A number may be either the perfect or imperfect power of a required root. 25 is a perfect square, but an imperfect cube. The exact root of an imperfect power can not be extracted and is called a surd. Prime numbers are imperfect powers of all their roots, except the first.

SQUARE ROOT.

ART. 168. The Square Root of a number is a number which multiplied by itself will produce the given number. Thus the square root of 16 is 4, since 4×4=16.

The process of finding the square root of a number is best understood by observing the manner in which the square of a number is formed, and the relation which the orders of the square bear to those of the root.

The first nine numbers are:

1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 9, and their squares

1, 4, 9, 16, 25, 36, 49, 64, 81. From which it is seen that the square of any number composed of one order of figures, can not contain more than two orders.

Conversely, that the square root of any number composed of one or two orders is composed of but one order.

It will further be seen that the numbers in the second line above are the only perfect squares found below 100, and that

the square root of any number between any two of these consecutive perfect squares is between the two corresponding roots above. Thus, 75 is not a perfect square and its square root is between 8 and 9.

The first nine numbers expressed by tens are,

10,

20, 30, 40, 50, 60, 70, 80, 90, and their squares,

100, 400, 900, 1600, 2500, 3600, 4900, 6400, 8100. From which it is seen that the square of tens gives no order below hundreds or above thousands. In the same manner it may be shown that the square of any number must contain at least twice as many orders, less one, as the number squared. If the left hand figure of the number squared is more than three, the square will always contain just twice as many orders as the root. Thus, the square of 456 contains six orders.

Again, every number may be regarded as composed of tens and units. Thus, 65 is composed 6 tens and 5 units, that is 60+5=65; 365, of 36 tens and 5 units, that is 365=360+5. Hence (65)2=(60)3+2× 60 × 5+(5)2=3600+600+25=4225, and (365)'= (360)+2×360 × 5+(5)'=129600+3600+25= 133225.

In like manner it may be shown that the square of any number is equal to the square of the tens plus twice the product of tens by units plus the square of units.

The two principles, above, determine the process of extracting the square root of a number.

Ex. 1. What is the square root of 4225 ?

Operation. 4225 65 36 6x2=125)62 5

-

Explanation. Since 4225 is composed of four orders, its root will be composed of but two; and since the square of units is composed of units and tens, and the square of tens, of hundreds and thousands, we separate the number into periods of two figures each, by placing a dot over units and another over hundreds.

62 5

Now 42 must contain the square of the ten's figure of the root. The greatest perfect square in 42 is 36, the square root of which is 6. Hence 6 is the ten's figure of the root. Sub

tracting the square of the ten's figure of the root from 42 hundreds, we have 6 hundreds for a remainder, to which, if the 25 units be added, we shall have 625, which is composed of twice the product of the tens of the root by the units (to be found) plus the square of the units.

Now the product of tens by units gives no order below tens, hence 62 tens must contain twice the product of the tens by the units. It may contain more, since the square of units may give tens.

If 62 tens be divided by 2 x 6 tens, or 12 tens, the quotient, 5, will be the unit figure of the root. By placing 5, the unit figure, at the right of 12 tens, and multiplying the result, 125, by 5, the product will be twice the tens by the units, plus the square of the units.

Ex. 2. What is the square root of 133225 ?

1. Separate the given number into periods of two figures each, commencing at units.

2. Find the greatest perfect square in the left hand period and place its root on the right as the highest order of the root. 3. Subtract the square of the root figure from the left hand period, and to the remainder annex the next period for a dividend.

4. Double the part of the root already found for a trial divisor, and see how many times it is contained in the dividend, exclusive of the right hand figure, and write the quotient as the next divisor of the root, and also at the right of the trial divisor.

5. Multiply the divisor thus formed by the figure of the root last found, and subtract the product from the dividend.

6. To this remainder annex the next period for the next dividend, and divide the same by twice the root already found, and continue in this manner until all the periods are used.

Notes.-1. The left hand period often contains but one figure.

2. Twice the root already found is called the trial divisor, since the quotient may not be the next figure of the root. The quotient may be too large, in which case it must be made less. The true divisor is the trial divisor with the figure of the root found annexed.

3. When any dividend exclusive of its right hand figure is not large enough to contain its trial divisor, place a cipher for the next figure of the root, and double the root thus formed for a new trial divisor, and form a new dividend by bringing down the next period.

4. When there is a remainder after all the periods are used, annex a period of two ciphers, and thus continue the operation until the requisite number of decimal places is obtained. In this case, there will be a remainder, how far soever the operation be continued, since the square of no one of the nine digits ends with a cipher.

5. The square root of a common fraction may be found by taking the root of both terms, when they are perfect squares. When both terms of a fraction are not perfect squares, and can not be changed to perfect squares, the root of the fraction can not be exactly found. The approximate root, however, may be found by multiplying the numerator of the fraction by the denominator, and extracting the root of the product, and dividing the result by the denominator. By extracting the root to decimal places the error may be further lessened.

6. In finding the square root of a decimal or a mixed decimal, commence separating into periods at the order of units for the whole number, and at the order of tenths for the decimal. If there be an odd number of decimal places, annex a cipher.

7. Mixed numbers must first be reduced to improper fractions or to mixed decimals.

THE RIGHT-ANGLED TRIANGLE.

ART. 169. An angle is the divergence of two lines meeting at a common point,

Angles are divided into three classes; acute, obtuse, and right.

The annexed figures illustrate the three kinds of angles.

Acute.

Obtuse.

Right angle.

A triangle is a figure bounded by three straight lines. It also contains, as its name indicates, three angles.