3a √8a3×26 √4a3c=6ab √√/32a1c=12a2b*√2c. When the radicals have not a common index, they should be reduced to one. For example, 3a √bx5b √2c=15 ab√8b+c3. 6. Multiply 2, 3, and √5 together. 7 8. Multiply (4√+5V) by (√+2√). 9. Divide 11. Divide Va+√ō by ‘√a−√õ. Ans. Formation of Powers, and Extraction of Roots. 229. By raising a to the nth power, we have m ("/a)"="/a>"/axva...="√a", by the rule just given for the multiplication of radicals. Hence, for raising a radical to any power, we have the following RULE. Raise the quantity under the sign to the given power, and affect the result with the radical sign, having the primitive index. If it has a co-efficient, first raise it to the given power. Thus, *√4a3)2=*√(4a3)2=*√16ao=2a*√ a2 ; When the index of the radical is a multiple of the power, the result can be reduced. For, √2a√ √2a (Art. 214): hence, to square √2a, we have only to omit the first radical, which gives (‘√2a)2= √2a. 3√36. hence Again, to square V36, we have √36= Consequently, when the index of the radical is divisible by the exponent of the power, perform this division, leaving the quantity under the radical unchanged. To extract the root of a radical, multiply the index of the radical by the index of the root to be extracted, leaving the quantity under the sign unchanged. This rule is nothing more than the principle of Art. 176, enunciated in an inverse order. When the quantity under the radical is a perfect power, of the degree of either of the roots to be extracted, the result can be reduced. It is evident that Va="√va; because both expressions are 230. The rules just demonstrated for the calculus of radicals, principally depend upon the fact that the n root of the product of several factors is equal to the product of the n' roots of these fac tors; and the demonstration of this principle depends upon this: When the powers, of the same degree, of two expressions are equal, the expressions are also equal. Now this last proposition, which is true for absolute numbers, is not always true for algebraic expressions. To prove this, we will show that the same number can have more than one square root, cube root, fourth root, &c. For, denote the general expression of the square root of a by x, and the arithmetical value of it by p; we have the equation x2=a, or x2=p2, whence x=±p. Hence we see that the square of P, which is the root of a, will give a, whether its sign be + or -. In the second place, let x be the general expression of the cube root of a, and p the numerical value of this root; we have the equation xa, or ap3. This equation is satisfied by making x=p. Observing that the equation 3=p3 can be put under the form 3-p3-0, and that the expression -p3 is divisible by x-p, (Art. 59), which gives the exact quotient, x2+px+p2, the above equation can be transformed into (x−p) (x2+px+p2)=0. Now, every value of a which will satisfy this equation will satisfy the first equation. But this equation can be verified by suppos ing x-p=0, whence x=p; or by supposing Hence, the cube root of a, admits of three different algebraic values, viz. Again, resolve the equation *=p1, in which p denotes the arith. metical value of /a. This equation can be put under the form -p=0. Now this expression reduces to (x2-p2) (x2+p2). Hence the equation reduces to (a-p3) (x2+p2)=0, and can be satisfied by supposing x-p2=0, whence x=±p; or by suppos ing x2+p2=0, whence x=±√-p2=±p √−1. We therefore obtain four different algebraic expressions for the fourth root of a. For another example, resolve the equation which can be put under the form Now -p reduces to therefore the equation becomes But 3-p3-0, gives x=p3, 26-p=0. (ñ3—p3) (x3+p3), 2 And if in the equation +p3=0, we make p=-p', it becomes 23-p'3-0 from which we deduce x=p', and Therefore the value of x, in the equation x-p=0, and conse. quently the 6th root of a, admits of six values, p, ap, a'p, —p, -ap-a'p, by making We may then conclude from analogy, that x in every equation of the form a-a=0, or "-p"-0, is susceptible of m different va lues, that is, the mth root of a number admits of m different alge. braic values. 231. If in the preceding equations and the results corresponding to them, we suppose as a particular case a=1, whence p=1, we shall obtain the second, third, fourth, &c. roots of unity. Thus +1 and −1 are the two square roots of unity, because the equation x-1=0, gives x=1. |