restoring x in the place of u, we have x = { p± √9+}p2, an equation consisting of two terms, since the expression as it implies only known operations, to be performed on given quantities, must be regarded as representing known quantities. Designating the two values of this expression by a and a', we If the exponent m be even, instead of the two values given above, we shall have four, since each radical expression may take the sign; then and these four values will be real, if the quantities a and a' are positive. All the values of x may be comprehended under one formula, by indicating directly the root of the two members of the equa The following question produces an equation of this kind. 161. To resolve the number 6 into two such factors, that the sum of their cubes shall be 35. 6 Let x be one of these factors, the other will be; then taking 216 the sum of their cubes 3 and we have the equation 3 + x+21635 x3, · 35 x3 If we consider 3 as the unknown quantity, we obtain, by the rule given for equations of the second degree, (3) 216. By going through the numerical calculations, which are indicated, we find The first value gives second value presents x = √ 27 = 3, 3 x = 8 = 2. for the second factor or 2, while the or 3; we have, therefore, in the one case 3 and 2 for the factors sought, and in the other 2 and 3. These two solutions differ only in the order of the factors of the given number 6. 162. The equations, we have been considering, are also comprehended under the general law given in art. 159; for the m values of va, va are to be multiplied by the roots of unity belonging to the degree denoted by the exponent m. Applying what has been said to the equation, 163. THE great number of cases, in which no exact root can be found, and the length of the operation necessary for obtaining it by approximation, have led algebraists to endeavour to perform immediately upon the quantities subjected to the radical sign, the fundamental operations, intended to be performed upon their roots. In this way we simplify the expression as much as possible, and leave the extracting of the root, which is a more complicated process, to be performed last, when the quantities are reduced to the most simple state, which the nature of the question will allow. The addition and subtraction of dissimilar radical quantities can take place only by means of the signs + and For example, the sums and the differences can be expressed only under their present form. The same cannot be said of the expression because the radical quantities of which it is composed, become similar, when they are reduced to their more simple forms, according to the method explained in art. 130. First, we have 164. With respect to other operations the calculus of radical quantities depends upon the principle already referred to, namely; that a product, consisting of several factors, is raised to any power by raising each of the factors to this power. So also, by suppressing the radical sign, prefixed to a quantity, we raise this quantity to the power denoted by the exponent of this sign. 7 7 For example, a raised to the seventh power, is a simply, since this operation, being the reverse of that which is indicated by the sign, merely restores the quantity a to its original state. According to the principles here laid down, if, for example, in the expression Alg. 22 7 √ax 7 we suppress the radical signs, the result ab will be the seventh power of the above product; and taking the seventh root, we find This reasoning, which may be applied to all similar cases, shows, that in order to multiply two radical expressions of the same degree together, we must take the product of the quantities under the radical sign, observing to place it under a sign of the same degree. We have by this rule 3/2ab3 × 74/5 a3 b c = 21 √10 aa ba c = 4 √√√ a2 — b2 × √√ aa + b2 = 4 √√(a2 — b2) (a2 + b2) = να 165. As the seventh power of the expression, for example, is, it will be seen, by taking the seventh root of this last result, that Hence to divide a radical quantity by another of the same degree, we must take the quotient arising from the division of the quantities under the radical sign, recollecting to place it under a sign of the same degree. 166. It follows from the rule, given in art. 164, for the multiplication of radical quantities of the same degree, that to raise a radical quantity to any power whatever, we have only to raise to this power the quantity under the radical sign, observing that the result 5 must take the same sign; thus to raise ab, for example, to the third power is to take the product and as the radical signs are all of the same degree, the quantities to which they belong, are to be multiplied together, and the radical sign to be prefixed to the product, which gives 7 7 5 In the same manner a b3 raised to the fourth power, gives Vas b12, which may be reduced to 7 ababs, by resolving a b12 into a7 b7 x abs, and taking the root of the factor a b (130). It may be observed, that when the exponent belonging to the radical sign is divisible by that of the power to which the proposed quantity is to be raised, the operation is performed by dividing the first exponent by the second. For example, Indeed va denotes a quantity, which is six times a factor in a, 3 and the quantity va, which is obtained by dividing 6 by 2, being only three times a factor in a, is consequently equivalent to the product of two of the first factors, and is therefore the second power of one of these factors, or of a. 6 |