16. 2x+y+ ‡x*y2 134. Let a and cd represent any two radi cals, after having been reduced to a common index. The quotient of the first by the second may be represented as follows: Reduce the radicals to a common index; then divide the coefficient of the dividend by that of the divisor for a new coefficient, and the quantity under the radical sign in the dividend by that in the divisor for a new quantity under the radical sign, leaving the index un changed. By combining the above rule with that for the divi sion of polynomials, any complicated radical expression may be divided by another. 6. Divide x + √xy + y, by √x + √√/xy + √ÿ. The two results are identical; but the second one has been obtained by following the ordinary rule for exponents. Hence, we conclude that the operation for divi sion is the same whether the exponents are entire or fractional. 7. (16x-1)+(24+1). 3 Ans. 8x3 + 2x13y + ‡x3y2 + ¡y°3. REDUCTION OF RADICALS. The first form may be thus transformed, by multiplying both terms by √ √e; and the second, by multiplying both terms by √b+ √c, giving If only one term of the denominator is affected by a radical, the same rule will hold good, 7. (√α + x + √α − x) ÷ (√a + x − √α − x). α va Ans. + equations, it oftentimes square root of expres In some cases, 136. In the solution of certain becomes necessary to extract the sions of the form, a + √ō, and a this operation may be performed, in other cases it cannot be performed. To investigate a rule for determining when the operation can be performed, and the manner of performing it, assume Squaring both members of (1) and (2), a + √o b = x2 + 2xy + y2 (4.) Adding (3) and (4), and omitting the common fac Adding and subtracting (5) and (6), 2-2 = ・ ・ (0.) y2 = Va2 b (3.) 2 Now, if a2b is a perfect square, its root may be represented by c. Substituting in (7) and (8), and extracting the square root of each member of both equations (Axiom 5), we have, These values, substituted in (1) and (2), give, The square root of the given quantities may be ex tracted when a2 b is a perfect square, and the roots may be obtained by substitution in (9) and (10). EXAMPLES. 1. Required the square root of 14+6√√5=14+ 180 Here, a=14, b=180, and c=√196-180-4: hence, |