By Harley Flanders and Justin J. Price (Auth.)

Algebra and Trigonometry

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The second equality, says something new, that you can extract the n-th rootfirst,then take the m-th power, and you get the same thing. To prove it we show that (Va)m has the property that 30 1. BASIC ALGEBRA defines am/n ; its Az-th power is am : [(Vä)m]n = (Vä)mn = [(Vä)n]m = am. We prove Rule (2) by writing s and t with a common denominator and using rules for integer exponents and radicals. Thus we write m p /= -, n s = —, n «>0. Then _ asat am/nap/n _ ^ / ^ Γ ^ρ = ß(w+j>)/n = — yamap = ^(Wn)+(p/n) = V^m+P ^s+f Rule (3) is proved similarly.

I + 1 = *_±_L x x 9. 4x2 - Ix = 2x + 1 2. — L ^ = 2 x+1 4. x2 + 2x = x(x + 1) + x 6. x2(x - 3) = 0 8. VI + x2 = x + 3 10. (JC + l) 2 - x2 = 2(x + 1) - 1. Find all solutions: 11. 4x = 24 13. x2 + 4x + 4 = 9 12. ΐχ = 2 14. 3x + 7 = 17. Vx~^T = 1 19. (x2 - 4)(x2 - 9) = 0 18. 1/x2 = 9 20. x2 = x. 15. JC(S - 2)(JC - 7) = 0 16. 4(JC + 1) 2x2 + 1 = 3JC2 - 8 - x 2. LINEAR EQUATIONS In this section and the next, we study two of the simplest and most common types of equations, linear and quadratic equations.

If so, of what? 31. x2 + 16x + 64 32. y4 + 2y2 + 1 33. z 6 + 4z3 + 4 34. I602 + 80 + 1 35. x2y2 + 3x^ + 9 36. 9c2 - 30a/ + 25d2 37. (x2 + l) 2 + 2(x2 + 1) + 1 38. 9u2 + 9w + 1 39. a2b8 - 4ab4c2 + 4c4 40*. x2 + 2xy + y2 + 4xz + 4yz + 4z2. Explain these party games: 41. Take a number from 1 to 10. Square the number one larger and square the number one smaller. Subtract. Divide by the number you started with. Now you have 4. 42. Multiply your number by the number 4 larger. Add 4. Take the square root.