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4a4 +12a3x +132?x2 +6ax3+x+(2a2 +3ax+x2

4a4. 499 +3ax)12a3x+13a? x2

12a8% t-9a2x2

4a2 +6ax+x2) 4a xo +6ax3+x*

4a2x2 +6ax3 + x4

*

Note. When the quantity to be extracted has no exact root, the operation may be carried on as far as is thought necessary, or till the regularity of the terms shows the law by which the series would be continued.

EXAMPLE.

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1. It is required to extract the square root of 1 + x.

X2

3
5x 4

&c.
2 8 16 128
1

1+1(1+

+

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Here, if the numerators and denominators of the two last terms be each multiplied by 3, which will not alter their values, the root will become X2 3x3 3.5x4 3.5.725

+

&c. 2 2.4 ' 2.4.6 2.4.6.8 2.4.6.8.10 where the law of the series is manifest.

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1+

+

EXAMPLES FOR PRACTICE.

16

2. It is required to find the square root of a4 +42*x + 6a? x2 +4ar3+x4

3. It is required to find the square root of x4 – 2x3 + 3

1 -x2 - 4x+

4. It is required to find the square root of 4x6 – 4x4 + 13x2 - 6x +9

6. Required the square root of 2® +425 +10x4 +20x3+ 25x2 + 24x+16.

6. It is required to extract the square root of a? +b.

7. It is required to extract the square root of 2, or of 1+1.

CASE III.

To find any root of a compound quantity.

RULE.

Find the root of the first term, which place in the quotient; and having subtracted its corresponding power from that term, bring down the second term for a diyidend.

Divide this by twice the part of the root above determined, for the square root; by three times the square of it, for the cube root, and so on ; and the quotient will be the next term of the root

Involve the whole of the root, thus found, to its proper power, which subtract from the given quantity, and

divide the first term of the remainder by the same divisor as before ; and proceed in this manner till the whole is. finished. (p)

EXAMPLES

1. Required the square root of at-2a3x+3a% 38 -- 20x3 +x4.

2a3x+3ax?:- 20x3 +24(as--axt.x2

a 4 an

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(0) As this rule, in high powers, is often found to be very laborious, it may be proper to observe, that the roots of various com-pound quantities may sometimes be easily discovered, as follows:

Extract the roots of all the simple terms, and connect them together by the signs + or -, as may be judged most suitable for the purpose ; then involve the compound root, thus found, to its proper power, and if it be the same with the given quantity, it is the root required. But if it be found to differ only in some of the signs, change them from t to ry

or from marco to + till its power agrees with the given one throughout. ,

Thus, in the third example next following the root is 2a --3x, which is the difference of the roots of the first and last terms: and in the fourth example, the root is a+b+c, which is the sum of the roots of the first, fourth, and sixth terms. The same may Iso be observed of the sixth example, where the root is found from the first and last terms.

F

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2. Required the cube root of 6 + 6x5 - 40x3 +96464.

26 +6:05 - 40x3 +963-64(x2+2x–-4

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3x 4 )6x5

26 +6x5 + 12x4 +82 3

3x4) – 12.04

28 +6x5 -- 40x3 +963-64

3. Required the square root of 4a3 - 12ax+9.zo.

4. Required the square root of a3 +2ab+2ac+b3+ 2bc+ca.

5. Required the cube root of x6 65 +1524 - 20x3 + 15x2 - 6x+1.

6. Required the 4th root of 16a+ -96a31+216ax 2160x3 +81x4.

7. Required the 5th root of 3225-80x4 +80x3 - 40:x2 + 10x -1.

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OF IRRATIONAL QUANTITIES,

OR SURDS.

IRRATIONAL quantities, or surds, are such as have no exact root, being usually expressed by means of the radical sign, or by fractional indices ; in which latter case,

i

the numerator shows the power the quantity is to be raised to, and the denominator its root.

Thus, /2, or

27, denotes the square root of 2 ; and Vap, or aš, is the square of the cube root of a, &c. (9)

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CASE I.

To reduce a rational quantity to the form of a surd.

RULE.

Raise the quantity to a power corresponding with that denoted by the index of the surd ; and over this new quantity place the radical sign, or proper index, and it will be of the form required.

EXAMPLES.

6

1. Let 3 be reduced to the form of the square root.

Here 3X3=32=9; whence ✓9 Ans. (2. Reduce 2x3 to the form of the cube root.

Here (2x2)3=880 ; whence V8x0, or (8x®)Ans. . 3. Let 5 be reduced to the form of the square root. 4. Let – 3x be reduced to the form of the cube root. 5. Let - 2a be reduced to the form of the fourth root.

6. Let az be reduced to the form of the fifth root, and Vatro, a and va

to the form of the square root. 2a ba Note. Any rational quantity may be reduced by the above rule, to the form of the surd to which it is joined, and their product be then placed under the same index, or radical sign.

a

(7) A quantity of the kind here mentioned, as for instance ✓ 2, is calles an irrational number, or a surd, because no number, either whole or fractional, can be found, wbich when multipled by itself, will produce But its approximate value may be jetermined to any degree of exactness, by the common rule for extracting the square root, being 1 and certain non periodic deci mals, which never terminate.

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