Algebra Practice Problems

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Simplify the following.

1. y = (x^{2})^{3} * x^{4}/(x^{10})

2. y = x^{1/2}* x^{5}

3. y = 2[(x-2)^{2} +6x]

4. z = (x+y)^{2} – (x-y)^{2}

5. y = 5a(a^{5})^{2}/(a^{6})^{2}

6. z = xy^{ -2}/x^{ -2}y / [x^{ -2}y^{3}/xy^{ -1}]

7. x/(x-1) + 2/x =0

Solve each of the following for x, so that x is on the left hand side of the equation and everything else is on the right hand side.

8. xy = 10

9. 2xy + b = 4

10. x^{2}/6 – y^{2}/10 – 5 = 0

11. e^{x} = 5

12. 8^{x} = 15

13. 0.10 = ln x

14. (x+a)/b +b = a^{2}

15. xyz – z = y^{2}

16. x^{1/2}/6 = 1

17. Solve for x and y: 9(x+1) + 5(y+3) = 5

7(x-3) – 3(2y+1)= -19

Use factoring to solve these problems:

18. x^{2} – 5x + 6 = 0

19. x^{4} – 2x^{2} = 0

20. x^{3}e^{x} – 27e^{x} = 0

21. x^{2} +98x = 200

22. This one is already factored. What are the solutions?

e^{x}(x^{2}-1)/[e^{2x}sin^{2}x – cos(2x)] = 0

Solutions

1. y = (x^{2})^{3} * x^{4}/(x^{10})

y = 1

2. y = x^{1/2}* x^{5}

y = x^{11/2}

3. y = 2[(x-2)^{2} +6x]

y = 2x^{2} – 4x +32

4. z = (x+y)^{2} – (x-y)^{2}

z = 4xy

5. y = 5a(a^{5})^{2}/(a^{6})^{2}

y = 5/a

6. z = xy^{ -2}/x^{ -2}y / [x^{ -2}y^{3}/xy^{ -1}]

z = x^{6}/y^{7}

7. x/(x-1) + 2/x =0

Multiply the first term by x/x and the second term by (x-1)/(x-1). This is the same as multiplying by 1, so it doesn