How to multiply complex numbers - ACT Math
Card 0 of 9
The solution of 
is the set of all real numbers 
such that:
The solution of
Square both sides of the equation: 
Then Solve for x: 
Therefore, 
Square both sides of the equation:
Then Solve for x:
Therefore,
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What is the product of 
and 
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 
in the them you can simplify the 
to negative 1. Foil is first, outside, inside, last
First
Outside:
Inside
Last
Add them all up and you get 
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
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Which of the following is equal to 
?
Which of the following is equal to
Remember that since 
, you know that 
is 
. Therefore, 
is 
or 
. This makes our question very easy.
is the same as 
or 
Thus, we know that 
is the same as 
or 
.
Remember that since
Thus, we know that
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Complex numbers take the form 
, where 
is the real term in the complex number and 
is the nonreal (imaginary) term in the complex number.
Distribute: 
Complex numbers take the form
Distribute:
This equation can be solved very similarly to a binomial like 
. Distribution takes place into both the real and nonreal terms inside the complex number, where applicable.
This equation can be solved very similarly to a binomial like
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Complex numbers take the form 
, where 
is the real term in the complex number and 
is the nonreal (imaginary) term in the complex number.
Distribute and solve: 
Complex numbers take the form
Distribute and solve:
This problem can be solved very similarly to a binomial like 
.
This problem can be solved very similarly to a binomial like
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Complex numbers take the form 
, where 
is the real term in the complex number and 
is the nonreal (imaginary) term in the complex number.
Which of the following is equivalent to 
?
Complex numbers take the form
Which of the following is equivalent to
When dealing with complex numbers, remember that 
.
If we square 
, we thus get 
.
Yet another exponent gives us 
OR 
.
But when we hit 
, we discover that 
Thus, we have a repeating pattern with powers of 
, with every 4 exponents repeating the pattern. This means any power of 
evenly divisible by 4 will equal 1, any power of 
divisible by 4 with a remainder of 1 will equal 
, and so on.
Thus, 
Since the remainder is 3, we know that 
.
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
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Complex numbers take the form 
, where 
is the real term in the complex number and 
is the nonreal (imaginary) term in the complex number.
Simplify the following expression, leaving no complex numbers in the denominator.
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 
, there exists a conjugate 
such that 
.
This can also be applied to complex conjugates, which will eliminate the nonreal portion entirely (since 
)!
Multiply both terms by the denominator's conjugate.
Simplify. Note 
.
FOIL the numerator.
Combine and simplify.
Simplify the fraction.
Thus, 
.
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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Simplify the following:
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 
. Therefore, 
is 
. Based on this, we can simplify further:
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
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Simplify the following:
Simplify the following:
Begin this problem by doing a basic FOIL, treating 
just like any other variable. Thus, you know:
Recall that since 
, 
. Therefore, you can simplify further:
Begin this problem by doing a basic FOIL, treating
Recall that since
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