Such quantities are called surds. Hence a surd is the indicated root of an imperfect power of the degree indicated. 138 e. For the rule to extract any root of a fraction, reverse the rule given in § 120 for finding any power of a fraction, and we have the rule: Extract the required root of the numerator, for a new numerator; and the required root of the denominator, for a new denominator. Exercises. Find the indicated roots of the following fractions: RADICALS. Transformation of Radicals. 139. Let a and b denote any two numbers, and p the product of their square roots. Then √ax √b=p Squaring both members, we have ax b = p2 (1) (2) Extracting the square root of both members of (2), we have Since the second members are the same in Equations (1) and (3), the first members are equal; that is, The square root of the product of two quantities is equal to the product of their square roots. Similarly, if n denotes any number, then n√a will denote the nth root; that is, any root of a. Let denote the products of the nth roots of a and b. Then Vax Vb = 1 Raising both members to the nth power, we have ax b = mn Extracting the nth root of both members of (2), we have Vab=1 (1) (2) (3) Since the second members are the same in Equations (1) and (3), the first members are equal; that is, The nth root of the product of two quantities is equal to the product of their nth roots. 140. Let a and b denote any two numbers, and tient of their square roots. Then Extracting the square root of both members of (2), we have Since the second members are the same in Equations (1) and (3), the first members are equal; that is, The square root of the quotient of two quantities is equal to the quotient of their square roots. Similarly, if r denotes the quotient of the nth roots of a and b, we have Raising both members to the nth power, we have a Extracting the nth root of both members of (2), we have na (1) (2) (3) Since the second members are the same in Equations (1) and (3), the first members are equal; that is, The nth root of the quotient of two quantities is equal to the quotient of their nth roots. These principles enable us to transform radical expressions, or to reduce them to simpler forms. Thus, the expression" Hence 49b1× 2a. 98 ab*= √98 ab√496' x 2a; and, by the principle of § 139, √496* x 2a = √496' x √2a=7b2√2a. In like manner, √45a2b3c2d = √9a2b2c2 x 5bd 10 =3abc√5 bd. √864 a2bc11 = √144 a2b1c1o × 6 bc = 12 ab2c3√6bc. The coefficient of a radical is the quantity without the sign. Thus, in the expressions the quantities 762, 3 abc, 12ab2c5, are coefficients of the radicals. 141. Hence, to simplify a radical of the second degree, we have the following rule: Resolve the expression under the radical sign into two factors, one of which shall be a perfect square. Extract the square root of the perfect square, and then multiply this root by the indicated square root of the remaining factor. In like manner, to simplify a radical of the nth degree, that is, of any degree, we have the rule: — Resolve the expression under the radical sign into two factors, one of which shall be a perfect nth power. Extract the nth root of the perfect nth power, and then multiply this root by the indicated nth root of the remaining factor. Exercises. Reduce the following radicals to their simplest form : 142. A coefficient, or a factor of a coefficient, may be carried under the radical sign, raising it to the proper power, by squaring it if the radical is of the second degree, by cubing it if the radical is of the third degree, by raising it to the |