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 42 × 2, we take the square root of the square root of the given number. Thus, 86030562916; 291654, Ans. 2. Required the 6th root of 117649. Since 62 × 3, we have √117649 = 343; 3437, Ans. 8. Required the 6th root of a12 + 6a1ob + 15a8b2 + 20ab3 + 15a4b+6a2b5 + b6. Ans. a2 + b. 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 are true when m and n are fractional. We will therefore place + I. To show that a x a3 = aa 3. 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 (a2® × ar); and since ps and qr are integral, this last result becomes II. To show that a÷a = a q By transformations similar to those just employed, we have hence, by equating the values of x in (1) and (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. EXAMPLES. 1. Multiply ab3 by a3¿1, and simplify the product. a‡bŝ × a‡b‡ = (a‡ × a3) × (b3 × b‡) = a3b = ab√a, Ans. 2. Simplify the expression, (2 × 2). X (x* × x3)‡ = (x18)* = x*, Ans. 3. Multiply xa — 3x3 + x‡ by x1 — 2x1 — 3. 3x + x1 2x + 6x4 — 2x1 — 3x1 — 3x2 + 9x1 — 3x4 xa −5x + 4xa + 7x3 — 3xa, Ans. 4. Divide x 5x + 7 V x2 − 5 √x − 6√x by V-2√x + 3Vx. OPERATION. x − 5√x + 7Vx − 5 √ x − 6 V x \ Vx2 - 2√x + 3√x --- Vx-3x-2, Ans. 6. Multiply a2 by ab‡. 7. Find the product of a‡, a1, a3, and a ̄a. 8. Divide ac* by act. 11. Multiply 2x2 + √xy by 3√x — √xy. Ans. að — at Ans. 6x + 3√xy3 — 2√x1y3 — xy. Ans. aa— 3 +3a ̄† — a 14. Divide a 2a + at by a — 1. 15. Multiply a4 Ans. Va√b. Ans. at — at. a§+a2b*}+a3⁄4b§+ab2+a‡b§+630 16. Divide x3 + x3a3 + a‡ by x Ans. a3 — Z4. Ans. (√a + √b)(√a −√ō) (Va + Vi)(Va – 4b) /5 Ans. ¿(2√5 – 5√/2)a. Ans. √a + √b. 23. Simplify (√5 + 2)(√5 + √2)(V5 − √2) {} ((√13 + 3) (√13 + √/3)(√/13 √3)(√13 - √3) Ans.. |