CASE I. 258. To involve a radical quantity to any power. 1. Raise (ab) to the 2d power. From (255) we have {(ab)}}" = (ab)3 = (ab)3, Ans. 2. Raise V2ax to the 4th power. From (255) we have (V2ax)=√16a*x*. But since 62X3, we have, from (256), In practice, the simplification may be effected by canceling a factor from the index of the radical, and extracting the corresponding root of the quantity under the radical sign. Thus, in general terms, we have Hence the following "Varva. RULE. I. If the quantity is affected by a fractional exponent, multiply this exponent by the index of the required power. II. If the quantity is affected by the radical sign, raise the quantity under the radical sign to the required power; and if the result is a perfect power, of a degree corresponding to any factor of the radical index, cancel this factor from the index, and extract the corresponding root of the quantity under the radical sign. NOTE.-The coefficient may be involved separately. EXAMPLES FOR PRACTICE. 1. Raise 2a to the 3d power. Ans. a 8a. 2. Raise x'y' to the 2d power. Ans. xyxy. Ans. 162a2ac1. Ans. (a—b)3, 7. Raise axVax to the 4th power. Ans. 2bV/3a. Ans. (a—x) c3 (a −x). Ans. a x®. 8. Raise Ŵx3y3—x'y3 to the 2d power. Ans. xyxy(x—y)3. 9. Raise (a+x) to the 6th power. a 10. Raise 96cx to the 2d power. Ans. a+2ax+x". 2a Ans. V3cx. CASE II. 259. To extract any root of a radical quantity. 1. What is the square root of 49ax* ? Since the coefficient is a perfect square, 419a2x2 = 29a*x*. But from (257) we have 29a1x* = 23 9ax= 23ax3, Ans. 2. What is the 6th root of 5cd5c? Passing the coefficient under the radical sign, we have But by (256), 5cd3V5c = √125c3d®. 6 √125c'd V125c'd. = Reducing this result by canceling the factor 3 from the radical index, and taking the cube root of the quantity, we have RULE. I. If the quantity is affected by a fractional exponent, divide this exponent by the index of the required root. II. If the quantity is affected by the radical sign, extract the required root of the quantity under the radical sign, if possible; otherwise, multiply the index of the radical by the index of the required root, and simplify the result as in Case I. III. If the given radical has a coefficient, extract its root separately when possible; otherwise, pass the coefficients under the radical. 260. The principle established in (256), viz., that may be conveniently applied to the extraction of the higher roots of quantities, when the index of the required root is a composite number. EXAMPLES. 1. Required the 4th root of 8603056. Since 4 = 2×2, we take the square root of the square root of the given number. Thus, 86030562916; 2916 54, Ans. 2. Required the 6th root of 117649. Since 62X3, we have V117649343; =343; †343 = 7, Ans. 7. Required the 4th root of a-8ab+24a'l'-32ab'+166*. Ans. a-2b. Ans. a+b. 8. Required the 6th root of a"+6a1b+15a*b*+20a*b*+15a*b* +6a2b°+bo. GENERAL THEORY OF EXPONENTS. 261. It has already been shown that amxan = am+n, am an ~”, and (aTM)" =a””, m and n being integers, and either positive or negative. To prove that the above relations are true universally, it remains only to show that they hold true when m and n are fractional. + I. To show that a2 xa=a2 Reducing the exponents to a common denominator, we have But from the nature of fractional exponents, (222), the second member of this equation may be written and as the two factors have the same radical index (227) the result reduces to and since ps and qr are integral, this last result becomes II. To show that a÷a = a2 By transformations similar to those just employed, we have hence, by equating values of x in (1) and (5), (4) (5 We conclude, therefore, that in multiplication, division, involution and evolution, the same rule will apply, whether the exponents are positive or negative, integral or fractional |