Name                          .

Math 152B Practice Midterm II

Please do all of the following problems.  All work and all answers must be done on your own paper.  Credit earned will be based on the steps that you show that lead to the final solution.  Good Luck!

Problem 1   Find the root: 

Solution

        -2, since (-2)³ = -8

Problem 2   Simplify the expression.  Leave your answer in exponential form with only positive exponents.

 Solution

Problem 3  Multiply and Simplify

Solution

Problem 4   Simplify the expression.  Assume all variables represent positive real number.

 Solution

Problem 5    Multiply and Simplify

   Solution

Problem 6  Rationalize the denominator

             5
                       
          4 -

 Solution

                     5    (4 + )
                                                
                 ( 4 - )(4 + )

                      20 + 5
             =                          
                        16  -  6

                      20 + 5
             =                          
                            10

Problem 7 Solve

 Solution

        Square both sides:

                a² + 24  =  (a - 2)²

                a² + 24  =  a² - 4a + 4

                24  =  -4a + 4

                4a  =  -20

                a  =  -5

            Notice that when we plug x  =  -5 into the right hand side of the original equation, we get -7, while the left hand side produces 1.  Hence there is no solution.

Problem 8   Simplify

 Solution

Problem 9   Divide and write your answer in the form a + bi.

        2 + 5i
                   
        4 - 3i

Solution

        (2 + 5i)(4 + 3i)
                                
        (4 - 3i)(4 + 3i)

         8 + 6i + 20i + 15i²
  =                                  
                16 + 9

         8 + 26i -16
  =                        
               25

          -8  + 26i              -8             26
  =                         =               +            i
               25                   25            25

Problem 10  Answer the following true or false.  Please explain your reasoning.

1. 
   
   
   
   Solution
   
   False,  for example if x = 1, then
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

2. If z is a complex number and z is its conjugate, then their product, z z , is a real positive number.
   
   
   Solution
   
   
   True,  (a + bi)(a - bi) = a² + b².