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# Various algegra problems

(see attached file for full description)

Simplifying the folllowing expressions
3. (sqrt(x^3) + 3x)^2

5. The sixth term in the expansion of (2x - y)^8 is
6. Factor completely: x^4 - 3x^3 + 8x - 24
10. Rationalize the denominator: (x - 2)/(x - sqrt(x + 2))

11. solve for x: (3x + 4)/2 = 1 - x/3
12. solve for x(x - 3) = 4

13. (3 - 2i)^(-1)

14. solve: x^2 + 2x = -5
17. Find the solution set: |(2x - 5)/3| > = 3
20. How many liters of 50% alcohol solution must be added to 20 liters of a mixture that is 25% alcohol to obtain a 30% alcohol solution?
21. solve for x: x^4 + 2x^2 - 63 = 0
22. Find the solution set of 16x^2 = x^4.
23. The domain of the function defined by f(x) = sqrt(2x -x^2) is
24. if f(x) = 3x^2 - 5 and g(x) = 4x + 1, then (fg)(-1) =
25. if f(x) =x /(1 -x), then f^(-1)(x) is
26. which of the following equations does not define a one-to-one function?
27. An equation of the line with x-intercept 2 and y-intercept -3 is
28. The vertex of the parabola y = -x^2 - 2x + 8 is
29. The center, C, and the radius r of the circle x^2 + y^2 -4x + 2y = 4 are
30. When 4 - 5x^2 + 2x^3 is divided by x - 2, the remainder is
31. Counting multiplicites, the polynomial equation 3 - 5x + 3x^3 + x^4 = 0 has how many solutions?
32. The roots of the polynomial equation 2x^4 - 3x^3 + 19x^2 - 27x + 9 = 0 are
36. A logarithmic form of 2^y = x^3 /4 is
39. The maxtir is a triangular form of the augmented matrix associated with a particular system of equations. The system of equations has:
40. The determinant of the matrix is:

#### Solution Preview

Please see the attached file for detailed solution.

1. (3)

Basic exponential function properties:

2. (4)

3. (2)

4.

So

5. Using the binomial theorem, the sixth term is

(2)

6. grouping and extracting the GCF

(2)

7. (4)

8. (5)

9. (5),
10. (1)

11. Multiply LCD, which is 6 to both sides:

12. (5), solving the equation by factoring:

13.
(1)

14.

(2)

15. (5)

However, x - 3 is in the denominator, so
Therefore, there is no solution.

16.
Square both ...

#### Solution Summary

The solution covers various algebra aspects such as simplifying exponents and radical expressing, solving equations, roots of polynomials, functions, and system of equations, etc.

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