Part 19

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.)

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).

10. Assuming that when an apple falls from a tree the distance (S meters) through which it falls in any time (t seconds) is given by the formula S = (1/2)gt^2 (where g = 9.8), find to two decimal places the time taken by an apple in falling 15 meters. (College Entrance Board.)

Excellent practice may be obtained by solving the ordinary formulas used in arithmetic, geometry, and physics orally, for each letter in turn.

ARITHMETIC

p = br i = prt a = p + prt

GEOMETRY