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4th, To find the fourth term. Multiply the exponent of a in the third term by the coefficient of the same; divide the product by 3, (the number of terms already found); the quotient is the coefficient of the next term ; then take off a power of a and bring on a power of b, for the fourth term of the power complete: Continue this process till all the powers of a are exhausted, and the power of b be equal to that of the power required.

EVOLUTION.

32. Evolution is the operation by which roots are discovered, and is always opposed to Involution.

Roots are quantities by whose successive multiplication given powers are produced.

33. The roots of simple quantities are extracted, by dividing the exponent of the power by the exponent of the root required. Thus, the cube root of a3 is a, of 860 is 26.

The reason of this is deduced from g 27. 34.

Rules for extracting roots of compound quantities are deduced from a review of the steps by which they are involved. Thus, the square of a +b is a2 +2ub+62 ; that is, the square of any two quantities is equal to the squares of each of the quantities, together with twice their product. See Euclid, Book 2d, proposition 4th. Therefore when a quantity is proposed, whose square root is a compound quantity, you are first to arrange the terms as taught in division, (§ 22.) Thus, Let the square root of a' +-2ab+b2 be required.

a2+2ab-:-6* (ab
a?

2a + b)2ab+12

2ab +62

The following TARLE exhibits the first nine powers of the 9 digits.

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We will now proceed to lay down some rules for extracting the roots of numbers.

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8th power

256 6561

65536 390625

1679616! 5764801

16777216

43046721

loth power

$12.19683 | 262144|1953125 | 10097695|40353607 | 134217728

19683 | 262 144 1953125 | 10077696 40353607 | 134217728 3874204891

Rules for extracting the square root.

I. Divide the given number into periods of two figures, reckoning from the unit's place.

II. Find the greatest root contained in the left hand period, and place it as the first figure of the root : Subtract its square from the said period, and to the remainder bring down the next period for a refolvend.

III. Double the first part of the root for the first part of the divisor, and enquire how often this part is contained in the resolvend, neglecting the right hand place; the quot gives the next figure of the root.

IV. Annex the quotient also to the divisor, and multiply this number by the quotient; subtract the product from the refolvend, and to the remainder, if any, bring down the next period for a new refolvend.

V. Use the last divisor for the first part of a new ane, doubling the right hand figure; then proceed as before.

Note. Every period gives a figure in the root.

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Here enquire for the greatest root contained in the first period 20, which is 4, then place it as the first figure of the root, and subtract its square (16) from 20; to the remainder 4 annex the next period 25 for a resolvend; then divide 42 (neglect

ing the 5) by twice 4 or 8, and place the quotiene 5 in the root; also annex it to 8, and multiply this 85 by 5, the last figure of the root. And subtract this product from the refolvend ; and since there is no remainder, 2025 is an exact square, of which 45 is the root.

Ex. JI. Required the square root of 58264.

58264(241.37
4

44)182

176

481)664

481

4823)18300

14469

48267)383100

337869

45231 rem.

37. If, after the given number is exhausted, there be a remainder, annex periods of cyphers thereto, and continue the operation till the decimal part of the root terminate, repeat or circulate, or till you think proper to limit it.

38. Rules for extracting the square root of vulgar fractions or

mixt numbers.

Extract the square root of the numerator and of the denominator for their respective terms of the root required. Thus the

25 5 square root of is

6

Or,

36

39. If the numerator and denominator of the fraction proposed be not complete powers, place the square root of their product over the given denominator, and reduce this new fraction to its lowest terms for the fractional root required. Thus,

goo 30 5

25 X36 = 900

and .36

36

as before. Or,

36

30 6

40. Reduce the fraction to a decimal, and extra& its square root. Thus,

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41. In mixt numbers it will be best to reduce the fractional part to a decimal, to which prefix the integral part, and extract the square root of the whole. Thus,

123=3i, or rather, V12.35=3.5

EXTRACTION OF THE CUBE Root.

Required the cube root of a3+32*b+3ab' +63.

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