4. Solve graphically and algebraically 3x + 7y = 5, 8x + 3y = -18. 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 Review the proofs, for positive integral exponents, of: I. a^m x a^n = a^(m + n). II. (a^m)/(a^n) = a^(m - n). III. (a^m)^n = a^(mn). IV. [a^(mn)]^(1/n) = a^m. V. [a/b]^n = (a^n)/(b^n). VI. (abc)^n = a^n b^n c^n. ~To find the meaning of a fractional exponent.~ Assume that Law I holds for _all_ exponents. If so, a^(2/3) . a^(2/3) . a^(2/3) = a^(6/3) = a^2. Hence, a^(2/3) is _one of the three equal factors_ (hence the cube root) of a^2. Therefore a^(2/3) = [a^2]^(1/3). In the same way, a^(4/5) . a^(4/5) . a^(4/5) . a^(4/5) . a^(4/5) = a^(20/5) = a^4.