9. Which is the greater, 3 or 5? 10. Arrange in order of magnitude √3, 3/4, and †7. MULTIPLICATION OF RADICALS. 240. 1. Multiply √6 by √15. By Art. 233, √6 x √15 = √6 x 15 √90= 3√10, Ans. 2. Multiply √2a by √3a2. = Reducing to equivalent radicals of the same degree, √2a = (2a)1 = (2a)* = √(2a)3 = √8a3 √3a2 = (3 a2) 13 = (3 a2) * = √ (3 a2)2 = √9a* Hence, √2ax √3a2 = √8a3 × √9a* = √/72a2 = a72a, Ans. RULE. Reduce the radicals to equivalent radicals of the same degree. Multiply together the quantities under the radical signs, and write the product under the common radical sign. Note. The result should be reduced to its simplest form. 11. 43 and 3 √2. 13. 3, 2, and 12. Vay, Vyz, and Vzx. 14. V2x, V3x, and 15. Multiply 2√3+3√2 by 3√3-√2. 2√3+3 √2 3√3- √2 18+9√6 -2√6-6 18+7√6-6=12+7√6, Ans. 5 1 3x2 Note. It should be remembered that to multiply a radical of the second degree by itself simply removes the radical sign; thus, √3 x√3 = 3. 16. Multiply 3 √x2+1+4x by 2 √x2+1 − x. 20. 2a-3b and 4 √a +√b. 21. √x-√y+√z and √x+√Y −√%. 22. √x+1−2 √x and 2√x+1+√x. 23. √2−√3+ √5 and √2 + √3 + √5. 24. 3/5 2/6 + √7 and 6 √5+ 4√6 −2 √7. 25. 8√3+10 √⁄2 − 3 √5 and 4 √3 − 5 √2 − √5. 31. (√x+1+√x − 1) (√x + 1 − √ x − 1). 32. (3√2x+5+ 2√3 x − 1) (3 √2 x + 5 −2√3 x − 1). Reduce the radicals to equivalent radicals of the same degree. Divide the quantities under the radical sign, and write the quotient under the common radical sign. EXAMPLES. 1. Divide 15 by √5. Reducing to equivalent radicals of the same degree, we 242. Any power or root of a radical may be found by using fractional exponents. 1. Raise 12 to the third power. (√/12)3 = (12*)3 = 12a = 12a = √/12 = 2√/3, Ans. 2. Raise 3/2 to the fourth power. (3/2)1 = (23)+ = 23 = 3/2 = 3/16 = 23/2, Ans. Note 1. The following rule for the involution of radicals is evident from the above: If possible, divide the index by the exponent of the required power; otherwise, raise the quantity under the radical sign to the required power. 12. Extract the cube root of √27x3. † (√27 x ̈3) = ( √ 27 x31) 3 = ( √ (3x) 3) 3 = [(3x)}]} = = (3x)2 = √3x, Ans. 13. Extract the square root of $6. √/ (3/6) = (63) 3 = 6* = &/6, Ans. Note 2. The following rule for the evolution of radicals is evident from the above: If possible, extract the required root of the quantity under the radical sign; otherwise, multiply the index of the radical by the index of the required root. If the radical has a coefficient which is not a perfect power of the same degree as the required root, it should be introduced under the radical sign before applying the rule. Thus, 21. √(√æ3y12). 15. √(√125). 18. √(√a+b). 16. √(√32). 19. √(√x2-2x+1). 22. √(4√2). TO REDUCE A FRACTION HAVING AN IRRATIONAL DENOMINATOR TO AN EQUIVALENT FRACTION WHOSE DENOMINATOR IS RATIONAL. CASE I. 243. When the denominator is a monomial. The reduction is effected by multiplying both terms by a radical of the same degree as the denominator, having such a quantity under the radical sign as will make the denominator of the resulting fraction rational. |