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60. To put the coefficient of a surd under the radical sign. RULE I. Reduce the coefficient to the form of the given surd, by the preceding rule. II. Multiply the quantities under the radical sign by the reduced coefficient, and the product will be the answer ".

6. Given 2 va, in order to put the coefficient under the radical sign. -

First reducing 2 to the form of the square root, we have 2= y/4, then multiplying a by 4, and placing the common radical sign over the product, we have yar for the answer.

7. Introduce the coefficients of 39 ya and 2" va–w under the radical sign.

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“v3 cy', and “va'x2+z-z. 61. To reduce dissimilar surds to equivalent ones, having a common indew. RULE I. Reduce the indices to fractions having a common

denominator, and place each new numerator over the quantity to which it belongs, for a new index.

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II. Write 1 over the common denominator, and place this fraction as an index over the given quantities, with their new indices; if those quantities are numbers, involve them to the power denoted by the new index, over which place the said fraction for an index P.

9. Reduce 5; and 25 to equivalent surds, having a common index.

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Erplanation. I first reduce 3 and 3 to a common denominator, I place the numerator 8 over the quantity 3, and the numerator 9 over 2; then putting 1 for a numerator over 12, I place this fraction over 3° and 29, making 3olor and 27+, ; I then involve 3 to the 8th power, and 2 to the 9th; over each of which I put the index or for the answers.

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12. Reduce 73 and 8% to a common index. Ans. 117649)* and 40964. 13. Reduce wo, y} and z} to a common index. Ans. *, yor, and zoo. 14. Reduce at and bi to a common index. Ans, ao, and b". 15. Reduce a-Fos and z* to a common index. Ans, a 4-4 air-F6 a” +4 awo-Fo); and 2.1%. 62. To reduce surds to equivalent ones having a given inder. RULE I. Divide the indices of the surds by the given index, and place the quotients each over its proper quantity, for a new index. II. Over the given quantities with their new indices, write the given index, the results will be the surds required ".

16. Reduce 2+ and 3% to equal surds, having the common index 4.

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Wherefore 2-lo-T); and 334–51.3, are the quantities re

quired.
Erplanation. *

I first divide 3 and ; each by 4, and the quotients are 2 and 3; these I place as new indices respectively over 2 and 3, and over these the given index ... I then involve 2 to the square, and 3 to the 4th power, by which the new index of the former is taken away, and that of the latter reduced to 4 ; over the results I place the given index + for the answer.

-T- -I- - - r 17. Reduce aii and zi to equal surds, having- for a common

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a Dividing the indices is equivalent to extracting the root denoted by the divisor; and placing the divisor as an index over the result is equivalent to involving the result to the power denoted by the divisor : wherefore since equal evolution and involution take place, the value of the given quantity is not altered by the transformation effected under this rule; this is evident from Ex. 16, where 2+ is evidently equal to , vas)" – a y2}= w/2* =T4, and 3% is in like manner evidently equal to a w/3}}* = . w/33=" ygī)}=81;]+, all which is sufficiently plain from Art. 60.

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18. Reduce 2% and 3% to equal surds, having the common

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1. , . 1 19. Reduce a +y and 3% to equal surds, having 2 for a com

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63. To reduce surds to their simplest terms.

Rule I. Divide the given surd by the greatest power (of the same name with it) that will divide it without remainder, and place the said power and the quotient, with the sign x between them, under the radical sign.

II. Extract the root of the fore-mentioned power, and place its root before the said quotient, with the proper radical sign between them ".

21. Reduce v32 to its simplest terms.
OPERATION.
v32= v16 x2=4y2, the answer.

Earplanation.

The greatest square that will divide 32 is 16, and the quotient is 2; the root of 16 is 4, therefore 4 V2 is the answer.

* In this rule the given surd is resolved into two factors, one of which is a power of the same name with the surd.

Now it is evidently the same thing to multiply the remaining factor by this power, both being under the radical sign, or to multiply the factor under the sign by the root of the power not under the sign; thus, Ex. 21. A/32 is evidently equal to v^16 × 2 = x/16 × v2, but v16=4, wherefore v16 x A/2=4 x A/2=4A/2, as in the example ; wherefore, the transformation which takes place under this rule does not alter the value of the given surd, which was to be shewn.

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64. To reduce a fractional surd to its equivalent integral one.

RULE I. Multiply the numerator under the radical sign by that power of its denominator, whose index is 1 less than the index of the surd.

II. Take the denominator away from under the radical sign, and divide the coefficient by it, and the surd part will be an integral quantity, which must be reduced to its simplest terms by the foregoing rule ".

* To prove the truth of this rule, let a * v. be a given surd, of which it is

b required to reduce the radical part y – to an integer, without altering the c

value of the given quantity. Now it is evident, that if both terms of a fraction be multiplied by the same quantity, (let that quantity be whatever it may,) the

value of the fraction is not altered; wherefore let both terms of the given surd bc. 2 be multiplied by co, and it will become a * v. ; now the denominator co is

a complete power of the same denomination with the surd, and therefore (Art. 63.) it may be taken away, provided its root c be made the divisor of the coefficient a ; (for dividing by c is the same as dividing by 3 yes; and dividing either one factor, the other factor, or the product, by the same or equal quantities, produces in each case the same result;) wherefore the given surd

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b value with the given surd a yT, which was to be shewn.

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