14. 2/(2^(1/2)) + 3^(1/2)); (a^(1/2) + b^(1/2))/(a^(1/2) - b^(1/2)); 3/(3 - 3^(1/2)). 15. [3^(1/2) + 2^(1/2)]/[6^(1/2) + 3^(1/2) - 2^(1/2)]. Review the method of finding the square root of a binomial surd. (By inspection preferably.) Then find square root of: 16. 5 + 2[6^(1/2)]. 17. 17 - 12[2^(1/2)]. 18. 7 - 33^(1/2). ~Reference:~ The chapter on Radicals in any algebra, beginning at Addition and Subtraction of Radicals. MISCELLANEOUS EXAMPLES, ALGEBRA TO QUADRATICS Results by inspection, examples 1-10. Divide: 1. (x^(5/17) + y^(5/17))/(x^(1/17) + y^(1/17)). 2. (x - y)/(x^(1/3) - y^(1/3)). 3. (m^2 + n^2)/(m^(2/3) + n^(2/3)). 4. (x - y^2)/(x^(1/3) - [y^2]^(1/3)). Multiply: 5. [a^(-3/4) + 2/(m^(1/2))]^2. 6. (K^(-2/7) - g^(-11/25))^2. 7. (r^(2s) + l^(-3m))(r^(2s) - l^(-3m)). 8. [a^(-2) + b^(-3) - 1/(c^2)]^2. 9. (3K^x + 4t^(-3))(3K^x - 7t^(-3)). 10. (2y^(2/7) - 40K^3)(3y^(2/7) + 55K^3). Factor: 11. x^(2/3) - 64. 12. y^(3/5) + 27. 13. b^(3/2) - 8m^(-1). 14. 3p - 8p^(1/2) - 35. Factor, using radicals instead of exponents: 15. 60 - 7[3b^(1/2)] - 6b. 16. 15m - 2[[mn]^(1/2)] - 24n. 17. a - b (factor as difference of two squares). 18. a - b (factor as difference of two cubes). 19. a - b (factor as difference of two fourth powers). 20. Find the H. C. F. and L. C. M. of x^2 + xy^(1/2) - 2y, 2x^2 + 5xy^(1/2) + 2y, 2x^2 - xy^(1/2) - y. 21. Solve (short method) (x - 7)/(x - 8) - (x - 8)/(x - 9) = (x - 4)/(x - 5) - (x - 5)/(x - 6). 22. Simplify (ab/c + bc/a + ca/b)/(a/bc + b/ca + c/ab) x [((a + b + c)^2)/(ab + bc + ca) - 2]. (_Princeton._) 1. Solve for p: 2^(p - 3) = 128. 2. Solve for t: t^(3/2) = -27. 3. Find the square root of 8114.4064. What, then, is the square root of .0081144064? of 811440.64? From any of the above can you determine the square root of .081144064? 4. The H. C. F. of two expressions is a(a - b), and their L. C. M. is a^2b(a + b)(a - b). If one expression is ab(a^2 - b^2), what is the other? 5. Solve (short method): 5/(7 - x) - [(2-1/4)x - 3]/4 - (x + 11)/8 + (11x + 5)/16 = 0. 6. Solve 2/m - 3/n + 10/p = -3, 4/m + 5/p + 6/n = 15, 1/m - 1/n + 5/p = -1/2. 7. Simplify 21[2/3]^(1/2) - 5[4/5]^(1/2) + 6[4-1/6]^(1/2) - 10[3-1/5]^(1/2) + (40/3)[11-1/4]^(1/2). 8. Does [16 x 25]^(1/2) = 4 x 5? Does [16 + 25]^(1/2) = 4 + 5? 9. Write the fraction 5/(4 + 2[3^(1/2)]) with rational denominator, and find its value correct to two decimal places. 10. Simplify [{([p + [p^2 - q]^(1/2)]/2)^(1/2) + ([p - [p^2 - q]^(1/2)]/2)^(1/2)}^2]/[p + q^(1/2)]. (_Princeton._)