place to the right of the number, two, four, six, .. ciphers, then extract the integral part of the root of the product, and divide the result by 10, 100, 1000.....

Hence, in order to obtain any required number of decimals in the root, we

must

Place on the right hand of the proposed number twice as many zeros as we wish to have decimal figures; extract the integral part of the root of this new number, and then mark off in the result the required number of decimal places.

EXAMPLES.

(1) Extract the square root of 3 to six places of decimals.

(2) Extract the square root of 5 to six places of decimals.

(3) Extract the square root of 12 to six places of decimals.

Ans. 1.732050.

Ans. 2.236068.

Ans. 3.464101.

When half, or one more than half, the figures are found, the rest may be found by division.

(4) Extract the square root of 2 to nine places of decimals.

The first five figures of the root found by the ordinary method are 1.4142; with the remainder, 3836. The next divisor is 28284. Dividing 3836 by 28284, according to the ordinary method of division, produces 1356 for a quotient, which, annexed to 1.4142, before found, gives for the root required 1.41421356.*

Extract the square root of 11 to six places of decimals.

Ans. 3.316624.

EXTRACTION OF THE SQUARE ROOT OF FRACTIONS.

√a

We have seen (Art. 62) that = ; hence, in order to extract the square root of a fraction, it is sufficient to extract the square roots of the numnerator and denominator, and then divide the former result by the latter. This method may be employed with advantage when either one or both of the terms of the proposed fraction are perfect squares; but when this is not the case, it will be found inconvenient in practice. If, for example, we take the fraction

3, although

3
5

=

√3

√5

(since each of these expressions, when multiplied by it

self, produces the same quantity, 3), we must find an approximate value both for √3 and also for √5, and, after all, we shall not be able to determine at once the degree of approximation in the result. Under such circumstances the following process may be employed:

a

ab

; this

Let the proposed fraction be, this may be put under the form being premised, let r represent the integral part of the root of the numerator * The reason for this rule may be given thus: Let k be the part of the root already found, and ≈ the remaining part. Then k+z will be the whole root, and (k+z)2=k2+2kz +2 the given number; as z is but a small fraction of k, z2 will be a still smaller fraction, and may be neglected, so that the given number may, without sensible error, be considered equal to k2+2kz. But k2 has been taken away, and the remainder, 2kz, divided by 2k, gives z.

Make the denominator of the fraction a perfect square, by multiplying both terms of the fraction by the denominator; extract the integral part of the root of the numerator, and divide the result by the denominator.

Let it be required to extract the square root of

This fraction is the same as

9

square root of 91 is 9; hence is the root sought, a result within of the

true value.

91

13

A greater degree of approximation may, perhaps, be required. In this case, returning to the number extract the root of 91 to any required degree of approximation. Suppose, for example, we wish to find the root of 91 within

(13)

of the real value, it will become by (Art. 88) 91

1

9.53.... Hence

100

REMARK.-It frequently happens that the denominator of the fraction, although not a perfect square, has a perfect square for one of its factors, in which case the above operation may be simplified.

23 48

Let the fraction, for example, be 48 is equal to 16×3, or (4)3×3;

hence, multiplying both terms of the fraction by 3, it becomes

69

(12)2 ;

and the denominator is thus made a perfect square. Extracting the

In general, therefore, whenever the denominator of the fraction involves a factor which is a perfect square, multiply both terms of the fraction by the factor which is not a perfect square.

EXTRACTION OF THE SQUARE ROOT OF DECIMAL FRACTIONS.

90. This process is an immediate consequence of the preceding remark. Required, for example, the square root of 2.36.

This fraction is the same as

236
100

; in this case the denominator is a perfect

square; extracting, therefore, the integral part of the root of the numerator, we

Again, let it be required to extract the square root of 3.425.

This fraction is the same as

3425
1000

But 1000 is not a perfect square; it is,

however, equal to 100 × 10, or (10)3×10; thus, in order to render the denominator a perfect square, it is sufficient to multiply both terms of the frac

34250 34250

tion by 10, which gives

or

10000' (100)**

Extracting the integral part of tho

185

100'

root 34250, we find 185; hence the root required is or 1.85, a result

It appears from the above that the number of decimal places must always be made even before the operation commences.

If we wish to have a greater number of decimal places in the root, we must add on the right of 34250 twice as many zeros as we wish to have additional decimal figures.

From what has just been observed, we readily deduce for the extraction of the square root of a decimal fraction the following

RULE.

Annex ciphers till there are twice as many decimal places as are required in the root, and then proceed as in whole numbers; or, beginning at the decimal point, point off both ways the usual periods of two figures each.

EXTRACTION OF THE CUBE ROOT OF NUMBERS.

91. The numbers

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 100, when cubed, become

1000,

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1000000, 1000000000; and, reciprocally, the numbers in the first line are the cube roots of the numbers in the second.

Upon inspecting the two lines, we perceive that, among the numbers expressed by one, two, or three figures, there are only nine which are perfect cubes; consequently, the cube root of all the rest must be a whole number plus a fraction.

92. But we can prove, in the same manner as in the case of the square root, that the cube root of a whole number, which is not the perfect cube of some other whole number, can not be expressed by an exact fraction, and, consequently, its cube root is incommensurable with unity.

93. The difference between the cubes of two consecutive whole numbers is greater in proportion as the numbers themselves are greater; the expression for this difference can easily be found.

Let

a and a +1 be two consecutive whole numbers;

Then,

(a+1)3=a3+3a2+3a+1;

Hence,

(a+1)3-a3=3a2+3a+1;

that is to say, the difference of the cubes of two consecutive whole numbers is equal to three times the square of the less of the two numbers, plus three times the simple power of the number, plus unity.

Thus, the difference between the cube of 90 and the cube of 89 is equal to 3x (89)2+3x89+1=24031.

Let us now proceed to investigate a process for the extraction of the cube root of any number.

EXTRACTION OF THE CUBE ROOT.

94. The cube root of a proposed number, consisting of one, two, or three figures only, will be found immediately by inspecting the cubes of the first nine numbers in (Art. 91). Thus, the cube root of 125 is 5, and the cube root of 54 is 3 plus a fraction, for 3×3×3=27, and 4×4×4=64; therefore 3 is the approximate cube root of 54, within one unit of the true value.

For the purpose of investigating a new and simple rule for the extraction of the cube root, it will be necessary to attend to the composition of a complete power of the third degree. Now, since we have

(a+b)3=(a+b)(a+b)(a+b)=a3+3a2b+3ab2+b3,

it is obvious that the cube of a number, consisting of tens and units, will be algebraically indicated by the polynomial

a3+3a2b+3ab2+b3,

where a designates the number of tens, and b the number of units in the root sought. The number in the tens' place will evidently be found by extracting the cube root of the monomial a3, for Va3a, and removing a3 from the polynomial a3+3a2b+3ab2+b3, we have the remainder,

3a2b+3ab2+b3(3a2+3ab+b2)b;

and the difficulty that has been hitherto experienced in the extraction of the cube root entirely consists in the composition of the expression 3a2+3ab+b2, which is obviously the true divisor by which to divide the remainder, after subtracting a3, or the cube of the tens, for the determination of b, the figure of the root in the place of units. The part 3a2 of the expression 3a2+3ab+b2, being independent of b, the yet unknown part of the root, is employed as a trial divisor for the determination of b; but since the expression 3a2+3ab+b2 involves the unknown part of the root in its composition, it is obvious that the trial divisor 3a2, which does not contain b, will, at the first step of the operation, give no certain indication of the next figure of the root, unless the figure denoted by b be very small in comparison with that denoted by a; for the trial divisor 3a2 will be considerably augmented by the addend 3ab+b2 when b is a large number, while the augmentation, when b is a small number, will not so materially affect the trial divisor.

When the figure in the tens' place is a small number, as 1 or 2, it is hence obvious that little or no dependence can be placed on the trial divisor; but if a

be great and b small, the trial divisor, 3a2, will generally point out the value of b. All this will be evident if we consider that the relative values of a and b materially affect the true divisor, 3a2+3ab+b2. In the successive steps, however, of the cube root this uncertainty diminishes; for, conceiving a to designate a number consisting of tens and hundreds, and b the number o units, then the value of b being small in comparison with a, the amount of the effect of b in the addend 34b+b2 will be very inconsiderable; hence the trial divisor, 3a2, will generally indicate the next figure in the root.

To remove, in some measure, the difficulty which has hitherto been experienced in the extraction of the cube root, we shall proceed to point out two methods of composing the true divisor, 3a2+3ab+b2, and leave the student to select that which he conceives to possess the greater facility of operation.* 95. First method of composition of 3a2+3ab+b2.

Distinguishing the three columns from left to right by first, second, and third columns, we write a in the root, and also three times vertically in the first column; then a×a produces a2, which write, also, three times vertically in the second column; multiply the second a2 by a, placing the product, a3, under a3 in the third column; then, subtracting a3 from the proposed quantity, we have the remainder, 3a2b+3ab2+b3. The sum of the three quantities in the second column gives 3a for the trial divisor, by which find b, the next figure of the root, and to 3a, the sum of the last three written quantities in the first column, annex b; then the sum, 3a+b, is multiplied by b, and the product, 3ab+b2, is placed in the second column; then the trial divisor, 3a2, and the addend, 3ab+b2, being collected, give the true divisor, 3a2+3ab+b2, which multiply by b, and place the product, 3a2b+3ab2+b3, under the remainder, 3ab+3ab2+b3. When there is a remainder after this operation, the process may be continued by writing b twice in the first column, under 3a+b, and be once in the second column, under the last true divisor; then 3a +6ab3b, the sum of the last written three lines in the second column, will be another trial divisor, with which proceed as above. We have written a2 in the second column three times in succession, to assimilate the first step in the operation to the other successive steps, but the first trial divisor, 3a2, may be written at once, and the symmetry of the disposition of the quantities in the first steps disregarded.†

* These methods may be passed over by the student, as well as that given for the biquadrate root, and the method employed, which is described at (Art. 112), which is applicable to the extraction of the root of the third and fourth, as well as of any other degree. Three quantities are added each time; in the method on next page, two.