SAT Math : How to multiply complex numbers

Example Questions

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Example Question #1 : How To Multiply Complex Numbers

Find the product of (3 + 4i)(4 - 3i) given that i is the square root of negative one.

12 - 12i
7 + i
24
24 + 7i
0
Explanation:

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

Example Question #2 : How To Multiply Complex Numbers

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?

The correct answer is not listed.

Explanation:

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:

Example Question #3 : How To Multiply Complex Numbers

Simplify:

None of the other responses gives the correct answer.

Explanation:

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,

.

Example Question #4 : How To Multiply Complex Numbers

Multiply  by its complex conjugate.

None of the other responses gives the correct answer.

Explanation:

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.

Example Question #5 : How To Multiply Complex Numbers

Multiply  by its complex conjugate.

Explanation:

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.

Example Question #6 : How To Multiply Complex Numbers

What is the product of  and its complex conjugate?

The correct response is not among the other choices.

The correct response is not among the other choices.

Explanation:

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.

Example Question #7 : How To Multiply Complex Numbers

Multiply and simplify:

None of the other choices gives the correct response.

None of the other choices gives the correct response.

Explanation:

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:

Example Question #8 : How To Multiply Complex Numbers

Evaluate

Explanation:

is recognizable as the cube of the binomial . That is,

Therefore, setting  and  and evaluating:

.

Example Question #9 : How To Multiply Complex Numbers

Evaluate

None of the other choices gives the correct response.

Explanation:

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.

Example Question #10 : How To Multiply Complex Numbers

Raise  to the power of 3.

Explanation:

To raise any expression  to the third power, use the pattern

Setting :

Taking advantage of the Power of a Product Rule:

Since ,

and

:

Collecting real and imaginary terms:

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