ACT Math : Algebra
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Example Questions
Example Question #1 : How To Divide Complex Numbers
Simplify:
This problem can be solved in a way similar to other kinds of division problems (with binomials, for example). We need to get the imaginary number out of the denominator, so we will multiply the denominator by its conjugate and multiply the top by it as well to preserve the number's value.
Then, recall 
This is as far as we can simplify, so it is our final answer.
Example Question #12 : Complex Numbers
The solution of 
Square both sides of the equation: 
Then Solve for x:
Therefore,
Example Question #1 : How To Multiply Complex Numbers
What is the product of 
Multiplying complex numbers is like multiplying binomials, you have to use foil. The only difference is, when you multiply the two terms that have 
First
Outside:
Inside
Last
Add them all up and you get
Example Question #3 : How To Multiply Complex Numbers
Simplify the following:
Begin this problem by doing a basic FOIL, treating 
Recall that since 
Example Question #3 : How To Multiply Complex Numbers
Complex numbers take the form 
Distribute:
This equation can be solved very similarly to a binomial like 
Example Question #4 : How To Multiply Complex Numbers
Complex numbers take the form 
Distribute and solve:
This problem can be solved very similarly to a binomial like 
Example Question #22 : Complex Numbers
Complex numbers take the form 
Which of the following is equivalent to 
When dealing with complex numbers, remember that 
If we square 
Yet another exponent gives us 
But when we hit 
Thus, we have a repeating pattern with powers of 
Thus,
Since the remainder is 3, we know that 
Example Question #3 : How To Multiply Complex Numbers
Simplify the following:
Begin by treating this just like any normal case of FOIL. Notice that this is really the form of a difference of squares. Therefore, the distribution is very simple. Thus:
Now, recall that 
Example Question #8 : How To Multiply Complex Numbers
Which of the following is equal to 
Remember that since 
Thus, we know that 
Example Question #41 : Squaring / Square Roots / Radicals
Complex numbers take the form 
Simplify the following expression, leaving no complex numbers in the denominator.
Solving this problem requires eliminating the nonreal term of the denominator. Our best bet for this is to cancel the nonreal term out by using the conjugate of the denominator.
Remember that for all binomials 
This can also be applied to complex conjugates, which will eliminate the nonreal portion entirely (since 
Thus, 
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