If the index of the radical is a multiple of the exponent of the power, the operation may be simplified. Thus, († 2a)2=( √ √ 2a)2=√ 2a, (Art. 192.) mn n m m In general, ("Vα)^= ( ~√√√ā)"="ā. Hence, a If the index of the radical is divisible by the exponent of the power, we may perform this division, and leave the quantity under the radical sign unchanged. 81a481a1 Thus, to raise /3a to the 4th power, we have =√3a, or, dividing 8 by 4, we obtain at once √3a. tract the roots of radicals, we have the following Rule.-Multiply the index of the radical by the index of the root to be extracted, and leave the quantity under the radical sign unchanged. Thus, the square root of 2a is †2a=†/2a. If the radical has a coëfficient, its root must also be extracted. If the quantity under the radical is a perfect power of the same degree as the root to be extracted, the process may be simplified. 3 Thus, 8a is equal (Art. 192) to 8a3= †/2a. 5. The cube root of (m+n)√/m+n. Ans. Vab. Ans. 2a2/2c. Ans. 7a. Ans. 42a2. Ans. Vm+n. IMAGINARY, OR IMPOSSIBLE QUANTITIES. 209. An imaginary quantity (Arts. 182, 193) is an even root of a negative quantity. Thus, -a, and b, are imaginary quantities. The rules for the multiplication and division of radicals (Art. 205) require some modification when imaginary quantities are to be multiplied or divided. Thus, by the rule (Art. 205), va×√—a=√—a× —α== va2±a. But, since the square root of any quantity multiplied by the square root itself, must give the original quantity, (Art. 198,) therefore, a×√—a——a. 210. Every imaginary quantity may be resolved into two factors, one a real quantity, and the other the imaginary expression, √—1, or an expression containing it. This is evident, if we consider that every negative quantity may be regarded as the product of two factors, one of which is -1. Thus, -a=α-1, —b2—b2×-1, and so on. Hence, √—a2=√ a2×—1=√a2X√−1=±a√−1. Since the square root of any quantity, multiplied by the square root itself, must give the original quantity; Therefore, (-1)2= √—1×√1 -1. (-1)=(y-1)2(-1)2=(-1)(−1)=+1. Attention to this principle will render all the algebraic operations, with imaginary quantities, easily performed. Thus, √-a × √-b=√@× √−1×√b× √−1 =√abx (-1)2=-yab. If it be required to find the product of a+by-1 by a-by-1, the operation is performed as in the margin. OPERATION. a+b√ −1 —ab√—1+b2 Since a2+b2=(a+b√—1) (a—by-1), any binomial whose terms are positive may be resolved into two factors, one of which is the sum and the other the difference of a real and an imaginary quantity. Thus, m+n=(√m+√ñ√=1){√m−√ñ√=1). 1. Multiply a2 by √—b3. 2. Find the 3d and 4th powers of a√—1. Ans. -ab. 6. Find the continued product of x+a, x+a√−1, 7. Of what number are 24+7-1, and 24-7√=1, the imaginary factors? Ans. 625. VI. THEORY OF FRACTIONAL EXPONENTS. 211. The rules for integral exponents in multiplication, division, involution, and evolution, (Arts. 56, 70, 172, and 194,) are equally applicable when the exponents are fractional. Fractional exponents have their origin (Art. 196) in the 4 extraction of roots, when the exponent of the power is not divisible by the index of the root. ทา Thus, the cube root of a2 is a3. So the nth root of am is a". m - a, and a, may be read a to the power of †, The forms a to the power of, and a to the power of minus MULTIPLICATION AND DIVISION OF QUANTITIES WITH FRACTIONAL EXPONENTS. 212. It has been shown (Art. 56) that the exponent of any letter in the product is equal to the sum of its exponents in the two factors. It will now be shown that the same rule applies when the exponents are fractional. 1. Let it be required to multiply a3 by a. But this result is the same as that obtained by adding the expo nents together. ! 2 15 22. Thus, a}_a}_a}+}_a}l + } } __a??. Hence, where the exponents of a quantity are fractional, To Multiply, Rule.-Add the exponents. 2. Let it be required to multiply a by a Adding —3 and §, we have. Hence, the product is a131⁄2, or 1ā. -- 213. By an explanation similar to that given in the preceding article, we derive the following rule. Where the exponents of a quantity are fractional, To Divide, Rule.-Subtract the exponent of the divisor from the exponent of the dividend. 6. 1 11 Ans. a 4 c Ans. (2) Ans. a-b. 1 Ans. x2y-y3. m+n (a+b)"×(a+b)1×(a—b)"\(a—b)3. Ans. (a2—b2)TM. POWERS AND ROOTS OF QUANTITIES WITH FRACTIONAL EXPONENTS. 214. Since the mth power of a quantity is the product of m factors, each equal to the quantity (Art. 172); 1 Therefore, to raise a" to the mth power, we have Hence, to raise a quantity affected with a fractional exponent to any power, Rule. Multiply the fractional exponent by the exponent of the power. 215. Conversely, to extract any root of a quantity affected by a fractional exponent, |