Imaginary Numbers - Algebra II
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Add:
Add:
When adding complex numbers, add the real parts and the imaginary parts separately to get another complex number in standard form.
Adding the real parts gives 
, and adding the imaginary parts gives 
.
When adding complex numbers, add the real parts and the imaginary parts separately to get another complex number in standard form.
Adding the real parts gives
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Subtract:
Subtract:
This is essentially the following expression after translation:
Now add the real parts together for a sum of 
, and add the imaginary parts for a sum of 
.
This is essentially the following expression after translation:
Now add the real parts together for a sum of
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Multiply:
Answer must be in standard form.
Multiply:
Answer must be in standard form.
The first step is to distribute which gives us:
which is in standard form.
The first step is to distribute which gives us:
which is in standard form.
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Divide: 
The answer must be in standard form.
Divide:
The answer must be in standard form.
Multiply both the numerator and the denominator by the conjugate of the denominator which is 
which results in
The numerator after simplification give us 
The denominator is equal to 
Hence, the final answer in standard form =
Multiply both the numerator and the denominator by the conjugate of the denominator which is
The numerator after simplification give us
The denominator is equal to
Hence, the final answer in standard form =
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Divide: 
Answer must be in standard form.
Divide:
Answer must be in standard form.
Multiply both the numerator and the denominator by the conjugate of the denominator which is 
resulting in
This is equal to 
Since 
you can make that substitution of 
in place of 
in both numerator and denominator, leaving:
When you then cancel the negatives in both numerator and denominator (remember that 
, simplifying each term), you're left with a denominator of 
and a numerator of 
, which equals 
.
Multiply both the numerator and the denominator by the conjugate of the denominator which is
This is equal to
Since 
When you then cancel the negatives in both numerator and denominator (remember that 
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What is the absolute value of 
What is the absolute value of
The absolute value is a measure of the distance of a point from the origin. Using the pythagorean distance formula to calculate this distance.
The absolute value is a measure of the distance of a point from the origin. Using the pythagorean distance formula to calculate this distance.
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Simplify the expression.
Simplify the expression.
Combine like terms. Treat 
as if it were any other variable.
Substitute to eliminate 
.
Simplify.
Combine like terms. Treat
Substitute to eliminate
Simplify.
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Consider the following definitions of imaginary numbers:
Then, 
Consider the following definitions of imaginary numbers:
Then,
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Find 
.
Find
Multiply the numerator and denominator by the numerator's complex conjugate.
Reduce/simplify.
Multiply the numerator and denominator by the numerator's complex conjugate.
Reduce/simplify.
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What is the value of 
?
What is the value of
Recall that the definition of imaginary numbers gives that 
and thus that 
. Therefore, we can use Exponent Rules to write 
Recall that the definition of imaginary numbers gives that
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What is the value of 
?
What is the value of
When dealing with imaginary numbers, we multiply by foiling as we do with binomials. When we do this we get the expression below:
Since we know that 
we get 
which gives us 
.
When dealing with imaginary numbers, we multiply by foiling as we do with binomials. When we do this we get the expression below:
Since we know that
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Evaluate: 
Evaluate:
Use the FOIL method to simplify. FOIL means to mulitply the first terms together, then multiply the outer terms together, then multiply the inner terms togethers, and lastly, mulitply the last terms together.
The imaginary 
is equal to:
Write the terms for 
.
Replace 
with the appropiate values and simplify.
Use the FOIL method to simplify. FOIL means to mulitply the first terms together, then multiply the outer terms together, then multiply the inner terms togethers, and lastly, mulitply the last terms together.
The imaginary
Write the terms for
Replace
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What is the value of 
, if 
=
?
What is the value of
We know that 
. Therefore, 
. Thus, every exponent of 
that is a multiple of 4 will yield the value of 
. This makes 
. Since 
, we know that 
.
We know that
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Combine like terms:
Distribute:
Combine like terms:
Combine like terms:
Distribute:
Combine like terms:
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Multiply: 
Multiply:
Use FOIL to multiply the two binomials.
Recall that FOIL stands for Firsts, Outers, Inners, and Lasts.
Remember that 
Use FOIL to multiply the two binomials.
Recall that FOIL stands for Firsts, Outers, Inners, and Lasts.
Remember that
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Rationalize the complex fraction: 
Rationalize the complex fraction:
To rationalize a complex fraction, multiply numerator and denominator by the conjugate of the denominator.
To rationalize a complex fraction, multiply numerator and denominator by the conjugate of the denominator.
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Multiply: 
Multiply:
Distribute:
combine like terms:
Distribute:
combine like terms:
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Simplify: 
Simplify:
Distribute the minus sign:
Combine like terms:
Distribute the minus sign:
Combine like terms:
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Simplify:
Simplify:
Use FOIL to multiply the binomials:
Change 
to -1:
Combine like terms:
Use FOIL to multiply the binomials:
Change
Combine like terms:
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