Practice Exam 1 -- Solutions
1. Since a power of a power corresponds to the product of exponents, and product of powers to the sum of exponents we obtain:
Hence, the answer is (C) .
2. We multiply numerator and denominator by , take into account that and obtain . We can cancel one y, and obtain (B) .
3. The expression can't be simplified. The radical can't be distributed over a sum! (C)
4. Can't be simplified! You can't cancel a's! (A)
5. By distributivity . That gives (A).
6. We can factor out x and expand from the formula . (C).
7. By expanding and writing the denominator as we obtain (A).
8. The radical of the radical corresponds to the power (1/2)*(1/2)= 1/4 . We obtain which gives us . This easily leads to (A) .
9. We have . Since the radical of a product is a product of radicals, and , we obtain (B).
10. The lowest common denominator is . Hence
We regroup terms in the numerator and obtain (A) .
11. A quotient is 0 if and only if the numerator is 0. Hence, the equation is equivalent to , which has two solutions x=4 and x=-4. At x=1 the expression is undefined. Thus x=1 is not a solution. The answer is (C) .
12. means by the geometric interpretation of the absolute value that the distance of x from 2 is less than 2. Hence, x must be between 0 and 4.
13. By simple algebra we obtain the solution .
14. We take the common denominator which is . We obtain which leads (A) .
15. Since the radical of a product is a product of radicals, and , , , we obtain (B) .
16. For the radical to be defined, it must be . Hence, x must be in the interval [-2,2].
17. , so f(10)=-99=f(-10). . Expanding the square and simplifying, we obtain .
18. Increasing in ( ) and ( ), decreasing in (-1,1).
19. Relative minimum at x=1, maximum at x=-1.
20. The function is odd as clearly , or, in other words, the graph is symmetric with respect to the origin.
21. No. It does not pass the vertical line test.
22. The function is . That is, . The graphs of g(x) and h(x) look as follows:
23. Solving for y gives us . Hence, the slope is .
24. The slope of a perpendicular line is the negative reciprocal of the slope of the original line. In our case, . Since the line passes through (0,0), the y intercept is 0. Hence, the equation is .
25. If you put all the parentheses correctly, you obtained the correct answer 32.0478....