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The same reasoning may be applied to all similar cases, as in the following example;

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167. If we reverse the methods given in the preceding article, we shall be furnished with rules for extracting the roots of radical quantities.

We perceive, by attending to the rule first stated, that if the exponents of the quantities under the radical sign are divisible by that of the root required, the operation may be performed as if there were no radical sign, only it is to be observed, that the result must be placed under the original sign.

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From the second rule given in the preceding article, it is evident, that the general method for finding the root of radical quantities, is to multiply the exponent belonging to the radical sign by that of the root, which is to be extracted.

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In fact, a is a quantity, which is five times a factor in a

5

(24, 129); but the cube root ofa, being also three times a factor in this last quantity, is found 5 x 3 times or 15 times a

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168. Since by multiplying the exponent of a quantity under a radical sign, by any number (166), we raise the root which is indicated, to the power denoted by this number, and by multiplying also the exponent belonging to the radical sign, by the same number (167), we obtain for the result a root of a degree equal to that of the power which was before formed, it is evident, that this second operation reduces the proposed quantity back to its original state.

35

The expression, as, for example, may be changed into a2 1, by multiplying the exponents 5 and 3 by 7; for multiplying the exponent of a3 by 7, we have, making use of the radical sign,

5

a21, the seventh power of the proposed radical quantity, and multiplying by 7 the exponent 5 belonging to the radical sign in the expression a21, we obtain the seventh root of the former result; this last process, therefore, restores the expression to its original value.

169. By this double operation, we reduce to the same degree any number of radical quantities of different degrees, by multiplying, at the same time, the exponent belonging to each radical sign, and those of the quantities under this sign, by the product of the exponents belonging to all the other radical signs. That the new exponents, which are thus found for the radical signs, are the same, is obvious at once, since they arise from the product of all the exponents belonging to the original radical signs; and after what has been said above, it is evident that the value of each radical quantity is the same as before.

By this rule we transform

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If we meet with numbers, under the radical signs, we shall be led, in applying this rule, to raise them to the power denoted by the product of the exponents belonging to the other radical signs.

170. In the same way, we may place under a radical sign a factor which is without one, by raising it to the power denoted by the exponent which accompanies this sign.

We may change, for example,

5

3

3

a into, and 2a into √/8a3 b.

171. After having, by the transformation explained above, reduced any radical quantities whatever, to the same degree, we may apply to them the rules, given in articles 164 and 165, for

the multiplication and division of radical quantities of the same degree.

Let there be the general expressions

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then by the rule given in art. 164, we have

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✔anp bng × √ bmr cms = √ amp bug+mr ¿ms,

for the product of the proposed radical quantities. We have also by the rule, art. 165,

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Remarks on some peculiar cases, which occur in the Calculus of Radical Quantities.

172. THE rules to which we have reduced the calculus of radical quantities, may be applied without difficulty, when the quantities employed are real. But they might lead the learner into error with regard to imaginary quantities, if they are not accompanied with some remarks upon the properties of equations with two terms.

For example, the rule laid down in art. 164, gives directly v=a x v=a=√ax-a√ a2;

and if we take a for vaa, we evidently come to an erroneous result, for the product a Xa, being the square of a, must be obtained by suppressing the radical sign, and is therefore equal to

α.

Bézout has obviated this difficulty, by observing, that when we do not know by what method the square a2 has been formed, we must assign for its root both a and—a; but when, by + means of steps already taken, we know which of these two quantities multiplied by itself produced a3, we are not allowed, in

seen,

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α.

going back to the root, to take the other quantity. This is evidently the case with respect to the expression √—a × √Fa; here we know, that the quantity aa, contained under the radical sign in the expression a, arises from a multiplied by — a; the ambiguity, therefore, is prevented, and it will be readily that in taking the root, we are limited to The difficulty above mentioned would present itself in regard to the product va X va, if we were not led, by the circumstance of there being no negative sign in the expression, to take immediately the positive value of a. In this case, since a2 arises from a multiplied by + a, its root must necessarily be + a.

There can be no doubt with respect to examples of the kind we have been considering; but there are cases, which can be clearly explained only by attending to the properties of equations with two terms.

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173. If, for example, it were required to find the product

1; reducing the second of these radical expressions to

the same degree with the first (169), we have

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a result which is real, although it appears evident, that the quantity a multiplied by the imaginary quantity ✔, ought to give an imaginary product. It must not be supposed, however, that the expression ✔ā is in all respects false, but only that it is to be taken in a very peculiar sense.

In fact, va, considered algebraically, being the expression for the unknown quantity x, in the equation with two terms,

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admits of four different values (159); for if we make a a, by taking a to represent the numerical value of Va, considered independently of its sign, or the arithmetical determination of this quantity, we have the four values

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α x + 1, αχ 1, α x + √=1, αχ the third of which is precisely the product proposed.

By a little attention, it will be readily perceived, whence the ambiguity of which we have been speaking, arises. The second power + 1 of the quantity 1 under the radical sign, as it may

arise as well from + 1 x + 1, as from -1 X1, causes the

quantity

m

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to have two values, which are not found in √—1. In general, the process by which the product × vō is formed, is reduced to that of raising this product to the power mn; for if we represent it by z, that is, if we make

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by raising the two members of this equation, first to the power m, we have

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This product, therefore, being determined only by means of its power of the degree mn, or by an equation of this degree with two terms, must have mn values (159). This will be per

m

να

n

ceived at once, if we reflect that the expressions and Võ, being nothing but the values of the unknown quantities x and y, in the equations with two terms,

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and, consequently, admitting of m and of n determinations, we have, by uniting the several m determinations of x, with the several n determinations of y, m n determinations of the product required.

When we are employed upon real quantities, there is no difficulty in finding the values, because the number of those, that are real, is never more than two (157), which differ only in the sign. 174. If we use the transformation explained in art. 159, the difficulty will be confined to the roots of +1 and -1; for if we make x at and y = ßu, a and 8 denoting the numerical values of va, vi considered without regard to the sign, the equations

m

n

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n

whence

xy=

m

✓ a × √ b = a 3 tu=aß

in which a represents the product of the numbers

m

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