text: # GRE Subject Test: Math : Imaginary Numbers & Complex Functions
  
  ## Example Questions
  
  ← Previous 1 3 4 5 6
  
  ### Example Question #241 : Gre Subject Test: Math
  
  Evaluate:
  
  Explanation:
  
  We can set  in the cube of a binomial pattern:
  
  ### Example Question #1 : Complex Imaginary Numbers
  
  Simplify the following product:
  
  Explanation:
  
  Multiply these complex numbers out in the typical way:
  
  and recall that  by definition. Then, grouping like terms we get
  
  ### Example Question #1 : Irrational Numbers
  
  Simplify:
  
  Explanation:
  
  Start by using FOIL. Which means to multiply the first terms together then the outer terms followed by the inner terms and lastly, the last terms.
  
  Remember that , so .
  
  Substitute in  for
  
  ### Example Question #1 : Imaginary Numbers & Complex Functions
  
  Simplify:
  
  Explanation:
  
  Start by using FOIL. Which means to multiply the first terms together then the outer terms followed by the inner terms and lastly, the last terms.
  
  Remember that , so .
  
  Substitute in  for
  
  ### Example Question #11 : Number Theory
  
  Simplify:
  
  Explanation:
  
  Start by using FOIL. Which means to multiply the first terms together then the outer terms followed by the inner terms and lastly, the last terms.
  
  Remember that , so .
  
  Substitute in  for
  
  ### Example Question #1 : Equations With Complex Numbers
  
  Solve for  and
  
  Explanation:
  
  Remember that
  
  So the powers of  are cyclic. This means that when we try to figure out the value of an exponent of , we can ignore all the powers that are multiples of  because they end up multiplying the end result by , and therefore do nothing.
  
  This means that
  
  Now, remembering the relationships of the exponents of , we can simplify this to:
  
  Because the elements on the left and right have to correspond (no mixing and matching!), we get the relationships:
  
  No matter how you solve it, you get the values .
  
  ### Example Question #2 : Imaginary Roots Of Negative Numbers
  
  Simplify:
  
  None of the Above
  
  Explanation:
  
  Step 1: Split the  into .
  
  Step 2: Recall that , so let's replace it.
  
  We now have: .
  
  Step 3: Simplify . To do this, we look at the number on the inside.
  
  .
  
  Step 4: Take the factorization of  and take out any pairs of numbers. For any pair of numbers that we find, we only take  of the numbers out.
  
  We have a pair of , so a  is outside the radical.
  We have another pair of , so one more three is put outside the radical.
  
  We need to multiply everything that we bring outside:
  
  Step 5: The  goes with the 9...
  
  Step 6: The last  after taking out pairs gets put back into a square root and is written right after the
  
  It will look something like this:
  
  ### Example Question #1 : Imaginary Numbers & Complex Functions
  
  Explanation:
  
  There are two ways to simplify this problem:
  
  Method 1:
  
  Method 2:
  
  Explanation:
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