Part 13

Review:

5. The squares of the numbers from 1 to 25.

6. The cubes of the numbers from 1 to 12.

7. The fourth powers of the numbers from 1 to 5.

8. The fifth powers of the numbers from 1 to 3.

9. The binomial theorem laws. (See Involution.)

Expand: (Indicate first, then reduce.)

10. (b + y)^7.

11. [(2a)/3 - 1]^5.

12. (x^2 + 2a)^5.

13. (x - y + 2z)^3.

14. A train lost one sixth of its passengers at the first stop, 25 at the second stop, 20% of the remainder at the third stop, three quarters of the remainder at the fourth stop; 25 remain. What was the original number? (M. I. T.)

~References:~ The chapter on Involution in any algebra. Also the references on the preceding page.

SQUARE ROOT

Find the square root of:

1. 1 + 16m^6 - 40m^4 + 10m - 8m^3 + 25m^2.

2. (a^2)/(x^2) + (6a)/x + 11 + (6x)/a + (x^2)/(a^2).

3. Find the square root to three terms of x^2 + 5.

4. Find the square root of 337,561.

5. Find the square root of 1823.29.

6. Find to four decimal places the square root of 1.672. (Princeton.)

7. Add 2/[(x - 1)^3] + 1/[(1 - x)^2] - 2/(1 - x) - 1/x.

8. Find the value of: (64^(1/3) . 12)/24 / 2 x 3 - (2 . 7^2)/(14) / 7 x 1 + (1^(1/3) . 1^7)/(1 . 1^2) - 4 . 0.

9. Simplify [(x + y)^5 + (x - y)^5][(x + y)^5 - (x - y)^5].

10. Solve by the short method: 5/(7 - x) - [(2-1/4)x - 3]/4 - (x + 11)/8 + (11x + 5)/16 = 0.

11. It takes 3/4 of a second for a ball to go from the pitcher to the catcher, and 1/2 of a second for the catcher to handle it and get off a throw to second base. It is 90 feet from first base to second, and 130 feet from the catcher's position to second. A runner stealing second has a start of 13 feet when the ball leaves the pitcher's hand, and beats the throw to the base by 1/8 of a second. The next time he tries it, he gets a start of only 3-1/2 feet, and is caught by 6 feet. What is his rate of running, and the velocity of the catcher's throw? (Cornell.)

~Reference:~ The chapter on Square Root in any algebra.

THEORY OF EXPONENTS