Complex Numbers - SAT Math
Card 0 of 344
Let
. What is the following equivalent to, in terms of
:

Let . What is the following equivalent to, in terms of
:
Solve for x first in terms of y, and plug back into the equation.

Then go back to the equation you are solving for:
substitute in

Solve for x first in terms of y, and plug back into the equation.
Then go back to the equation you are solving for:
substitute in
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For which of the following values of
is the value of
least?
For which of the following values of is the value of
least?
is the same as
, which means that the bigger the answer to
is, the smaller the fraction will be.
Therefore,
is the correct answer because
.
is the same as
, which means that the bigger the answer to
is, the smaller the fraction will be.
Therefore, is the correct answer because
.
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Define an operation
so that for any two complex numbers
and
:

Evaluate
.
Define an operation so that for any two complex numbers
and
:
Evaluate .
, so

Rationalize the denominator by multiplying both numerator and denominator by the complex conjugate of the latter, which is
:









, so
Rationalize the denominator by multiplying both numerator and denominator by the complex conjugate of the latter, which is :
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Simplify the expression by rationalizing the denominator, and write the result in standard form: 

Simplify the expression by rationalizing the denominator, and write the result in standard form:
Multiply both numerator and denominator by the complex conjugate of the denominator, which is
:







Multiply both numerator and denominator by the complex conjugate of the denominator, which is :
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Define an operation
so that for any two complex numbers
and
:

Evaluate 
Define an operation so that for any two complex numbers
and
:
Evaluate
, so

Rationalize the denominator by multiplying both numerator and denominator by the complex conjugate of the latter, which is
:








, so
Rationalize the denominator by multiplying both numerator and denominator by the complex conjugate of the latter, which is :
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Define an operation
such that, for any complex number
,

If
, then evaluate
.
Define an operation such that, for any complex number
,
If , then evaluate
.
, so

, so
, and

Rationalize the denominator by multiplying both numerator and denominator by the complex conjugate of the latter, which is
:







, so
, so
, and
Rationalize the denominator by multiplying both numerator and denominator by the complex conjugate of the latter, which is :
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Define an operation
such that for any complex number
,

If
, evaluate
.
Define an operation such that for any complex number
,
If , evaluate
.
First substitute our variable N in where ever there is an a.
Thus,
, becomes
.
Since
, substitute:

In order to solve for the variable we will need to isolate the variable on one side with all other constants on the other side. To do this, apply the oppisite operation to the function.
First subtract i from both sides.

Now divide by 2i on both sides.

From here multiply the numerator and denominator by i to further solve.



Recall that
by definition. Therefore,



.
First substitute our variable N in where ever there is an a.
Thus, , becomes
.
Since , substitute:
In order to solve for the variable we will need to isolate the variable on one side with all other constants on the other side. To do this, apply the oppisite operation to the function.
First subtract i from both sides.
Now divide by 2i on both sides.
From here multiply the numerator and denominator by i to further solve.
Recall that by definition. Therefore,
.
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Define an operation
as follows:
For any two complex numbers
and
,

Evaluate
.
Define an operation as follows:
For any two complex numbers and
,
Evaluate .
, so

We can simplify each expression separately by rationalizing the denominators.













Therefore,




, so
We can simplify each expression separately by rationalizing the denominators.
Therefore,
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From
, subtract its complex conjugate. What is the difference ?
From , subtract its complex conjugate. What is the difference ?
The complex conjugate of a complex number
is
, so
has
as its complex conjugate. Subtract the latter from the former:




The complex conjugate of a complex number is
, so
has
as its complex conjugate. Subtract the latter from the former:
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From
, subtract its complex conjugate.
From , subtract its complex conjugate.
The complex conjugate of a complex number
is
. Therefore, the complex conjugate of
is
; subtract the latter from the former by subtracting real parts and subtracting imaginary parts, as follows:




The complex conjugate of a complex number is
. Therefore, the complex conjugate of
is
; subtract the latter from the former by subtracting real parts and subtracting imaginary parts, as follows:
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From
, subtract its complex conjugate.
From , subtract its complex conjugate.
The complex conjugate of a complex number
is
. Therefore, the complex conjugate of
is
; subtract the latter from the former by subtracting real parts and subtracting imaginary parts, as follows:




The complex conjugate of a complex number is
. Therefore, the complex conjugate of
is
; subtract the latter from the former by subtracting real parts and subtracting imaginary parts, as follows:
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Find the product of (3 + 4i)(4 - 3i) given that i is the square root of negative one.
Find the product of (3 + 4i)(4 - 3i) given that i is the square root of negative one.
Distribute (3 + 4i)(4 - 3i)
3(4) + 3(-3i) + 4i(4) + 4i(-3i)
12 - 9i + 16i -12i2
12 + 7i - 12(-1)
12 + 7i + 12
24 + 7i
Distribute (3 + 4i)(4 - 3i)
3(4) + 3(-3i) + 4i(4) + 4i(-3i)
12 - 9i + 16i -12i2
12 + 7i - 12(-1)
12 + 7i + 12
24 + 7i
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has 4 roots, including the complex numbers. Take the product of
with each of these roots. Take the sum of these 4 results. Which of the following is equal to this sum?
has 4 roots, including the complex numbers. Take the product of
with each of these roots. Take the sum of these 4 results. Which of the following is equal to this sum?

This gives us roots of

The product of
with each of these gives us:




The sum of these 4 is:
![[(-2+5i) + (2-5i)] + [(5+2i) + (-5-2i)] =](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/26403/gif.latex)
![=[(-2)+2 +5i - 5i] + [5-5 + 2i - 2i] = [0] + [0] = 0](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/26404/gif.latex)
What we notice is that each of the roots has a negative. It thus makes sense that they will all cancel out. Rather than going through all the multiplication, we can instead look at the very beginning setup, which we can simplify using the distributive property:
![1(5+2i) + (-1)\cdot(5+2i)+ i(5+2i)+(-i) \cdot(5+2i) = (5+2i)\cdot[1+(-1)+i+(-i)]](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/26405/gif.latex)
![= (5+2i)\cdot[0] = 0](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/26406/gif.latex)
This gives us roots of
The product of with each of these gives us:
The sum of these 4 is:
What we notice is that each of the roots has a negative. It thus makes sense that they will all cancel out. Rather than going through all the multiplication, we can instead look at the very beginning setup, which we can simplify using the distributive property:
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Simplify:

Simplify:
Apply the Power of a Product Property:

A power of
can be found by dividing the exponent by 4 and noting the remainder. 6 divided by 4 is equal to 1, with remainder 2, so

Substituting,
.
Apply the Power of a Product Property:
A power of can be found by dividing the exponent by 4 and noting the remainder. 6 divided by 4 is equal to 1, with remainder 2, so
Substituting,
.
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Multiply
by its complex conjugate.
Multiply by its complex conjugate.
The complex conjugate of a complex number
is
. The product of the two is the number
.
Therefore, the product of
and its complex conjugate
can be found by setting
and
in this pattern:
,
the correct response.
The complex conjugate of a complex number is
. The product of the two is the number
.
Therefore, the product of and its complex conjugate
can be found by setting
and
in this pattern:
,
the correct response.
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Multiply
by its complex conjugate.
Multiply by its complex conjugate.
The complex conjugate of a complex number
is
. The product of the two is the number
.
Therefore, the product of
and its complex conjugate
can be found by setting
and
in this pattern:
,
the correct response.
The complex conjugate of a complex number is
. The product of the two is the number
.
Therefore, the product of and its complex conjugate
can be found by setting
and
in this pattern:
,
the correct response.
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Evaluate:

Evaluate:
is defined to be equal to
for any real or imaginary
and for any real
; therefore,

To evaluate a positive power of
, divide the power by 4 and note the remainder:

Therefore,

Substituting,

Rationalizing the denominator by multiplying both numerator and denominator by
:

is defined to be equal to
for any real or imaginary
and for any real
; therefore,
To evaluate a positive power of , divide the power by 4 and note the remainder:
Therefore,
Substituting,
Rationalizing the denominator by multiplying both numerator and denominator by :
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What is the product of
and its complex conjugate?
What is the product of and its complex conjugate?
The complex conjugate of a complex number
is
, so
has
as its complex conjugate.
The product of
and
is equal to
, so set
in this expression, and evaluate:
.
This is not among the given responses.
The complex conjugate of a complex number is
, so
has
as its complex conjugate.
The product of and
is equal to
, so set
in this expression, and evaluate:
.
This is not among the given responses.
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Multiply and simplify:

Multiply and simplify:
The two factors are both square roots of negative numbers, and are therefore imaginary. Write both in terms of
before multiplying:


Therefore, using the Product of Radicals rule:








The two factors are both square roots of negative numbers, and are therefore imaginary. Write both in terms of before multiplying:
Therefore, using the Product of Radicals rule:
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Evaluate 
Evaluate
is recognizable as the cube of the binomial
. That is,

Therefore, setting
and
and evaluating:
![x^{3} - 3x^{2} y + 3 xy^{2} - y^{3}=[ (7+4i) - (7-4 i) ]^{3}](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/817126/gif.latex)


Applying the Power of a Product Rule and the fact that
:
,
the correct value.
is recognizable as the cube of the binomial
. That is,
Therefore, setting and
and evaluating:
Applying the Power of a Product Rule and the fact that :
,
the correct value.
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