1. Rationalize the denominator of {6^(1/2) + 3^(1/2) - 3[2^(1/2)]}/{6^(1/2) - 3^(1/2) + 3[2^(1/2)]}. (_Univ. of Cal._) 2. Simplify [2^(n + 4) - 2(2^n)]/[2(2^(n + 3))]. (_Univ. of Penn._) 3. Find the value of [1 + 8^(-x/3)]/[(8x)^(1/2) + 10^(x - 2)], when x = 2. (_Cornell._) 4. Find the value of x if x^(6/5) = y^4, y^(2/3) = 9. (_M. I. T._) 5. A fisherman told a yarn about a fish he had caught. If the fish were half as long as he said it was, it would be 10 inches more than twice as long as it is. If it were 4 inches longer than it is, and he had further exaggerated its length by adding 4 inches, it would be 1/5 as long as he now said it was. How long is the fish, and how long did he first say it was? (_M. I. T._) 6. The force _P_ necessary to lift a weight _W_ by means of a certain machine is given by the formula P = a + bW, where _a_ and _b_ are constants depending on the amount of friction in the machine. If a force of 7 pounds will raise a weight of 20 pounds, and a force of 13 pounds will raise a weight of 50 pounds, what force is necessary to raise a weight of 40 pounds? (First determine the constants _a_ and _b_.) (_Harvard._) 7. Reduce to the simplest form: [[4/[2^(n + 2)]]^(1/n); [ax(a^(-1)x - ax^(-1))]/[x^(2/3) - a^(2/3)]. 8. Determine the H. C. F. and L. C. M. of (xy - y^2)^3 and y^3 - x^2y. (_College Entrance Board._) 1. Simplify (a - 8m)/(a^(1/3) - 2m^(1/3)) - 2a^(1/3)m^(1/3). 2. Simplify, writing the result with rational denominator: ([a^(1/2) + (1)/(x^(-1/2))]^2 - [(1)/(a^(-1/2)) - x^(1/2)]^2) / (x + [a^2 + x^2]^(1/2)). (_M. I. T._) 3. Find [7 - 48^(1/2)]^(1/2). 4. Expand ([a^3]^(1/2) - [b^5]^(1/2))^5. 5. Expand and simplify (1 - 2[3^(1/2)] + 3[2^(1/2)])^2. 6. Solve the simultaneous equations x ^(-1/2) + 2y^(-1/2) = 7/6, 2x^(-1/2) - y^(-1/2) = 2/3. (_Yale._) 7. Find to three places of decimals the value of {[(a + b)^(-1/3)]/[(11a + b^2)^(1/6)] . [({a^3 - b^3)^(-1/2)]/[(a - b)^(1/2)]}^(1/2), when a = 5 and b = 3. (_Columbia._) 8. Show that (10 - 4[5^(1/2)])/(5 + 3[5^(1/2)]) is the negative of the reciprocal of (10 + 4[5^(1/2)])/(5 - 3[5^(1/2)]). (_Columbia._) 9. Solve and check {5}/{[3x + 2]^(1/2)} = [3x + 2]^(1/2) + [3x - 1]^(1/2).